As to Jewish tradition, here are an interesting (imo) excerpt from the Babylonian Talmud, Baba Bathra, which reflects the considerable discussion about authorship and date (including one interesting opinion that the book of Job is a fiction):
The order of the Hagiographa is as follows: Ruth, Psalms, Job, Proverbs, Ecclesiastes, Song of Solomon, Lamentations, Daniel, Book of Esther, Book of Ezra, and Chronicles. And who wrote all the books? Moses wrote his book and a portion of Bil'am [Numbers, xxii.], and Job. Jehoshua wrote his book and the last eight verses of the Pentateuch beginning: "And Moses, the servant of the Lord, died." Samuel wrote his book, Judges, and Ruth. David wrote Psalms, with the assistance of ten elders, viz.: Adam the First, Malachi Zedek, Abraham, Moses, Hyman, Jeduthun, Asaph, and the three sons of Korach. Jeremiah wrote his book, Kings, and Lamentations. King Hezekiah and his company wrote Isaiah, Proverbs, Songs, and Ecclesiastes. The men of the great assembly wrote Ezekiel, the Twelve Prophets, Daniel, and the Book of Esther. Ezra wrote his book, and Chronicles--the order of all generations down to himself. [This may be a support to Rabh's theory, as to which, R. Jehudah said in his name, that Ezra had not ascended from Babylon to Palestine until he wrote his genealogy.] And who finished Ezra's book? Nehemiah ben Chachalyah.
There is a Boraitha in accordance with him who said that the last eight verses of the Torah were written by Joshua; namely: "It is written [Deut. xxxvi. 5]: 'And Moses the servant of the Lord died,' etc. Is it possible that Moses himself should have written 'and he died'? Therefore it must be said that up to this verse Moses wrote, and from this verse forward Joshua wrote. So said R. Joshua, according to others R. Nehemiah." Said R. Simeon to him: Is it possible that the Holy Scrolls should not have been complete to the last letter, and nevertheless it should read [ibid., xxxi. 26]: "Take this book of the law," etc. Therefore, we must say that up to this verse the Holy One, blessed be He, dictated, and Moses repeated and wrote it down; and from this verse forward He dictated, and Moses with tears in his eyes wrote it down; as thus it is read [Jer. xxxvi. 18]: "Then said Baruch unto them, With his mouth did he utter clearly all these words unto me, and I wrote them in the book with ink."
According to whom, then, is the following--that R. Joshua b. Aba, in the name of R. Gidel, quoting Rabh, said: "The last eight verses of the Pentateuch, when read from the Holy Scrolls, must be read by one person without any interruption"? Should it not be in accordance with R. Simeon? It may be also in accordance with R. Simeon; and the reason for the exception of these eight verses is because, as there was already a change at the writing by Moses (as said above), the change is made also here. "Joshua wrote his book"; but is it not written there: "And Joshua died"? This was written by Elazar. But is it not written there: "And Elazar died"? The book was finished by his son Pinchas.
"Samuel wrote his book." But is it not written: "And Samuel died"? The book was finished by Gad the seer and Nathan the prophet.
"David wrote the Psalms," etc. But why did the Boraitha not enumerate also Ethan the Ezrachite? Said Rabh: "The latter and Abraham are identical." It enumerates Moses, and also Hyman; did not Rabh say that by Hyman is meant Moses? There were two Hymans.
"Moses wrote his book," etc. This is a support to R. Levy b. Lachma, who said that Job lived in the time of Moses. Rabha, however, said: Job lived in the time of the spies which were sent by Moses to investigate Palestine.
One of the rabbis was sitting before R. Samuel b. Na'hmeni and said: Job never existed; and is mentioned in the Scripture only for an example. Said he to him: The Scripture is against your theory, as it states plainly [Job, i. 1]: "There was a man," etc. But according to your theory it is also written [II Sam. xii. 3]: "But the poor man had nothing," etc. Was it so in reality? It was written only for an example! The same may be said concerning Job? If it were so, why, then, his name and the name of the country he came from?
R. Johanan and R. Elazar both said that Job was among the ancestors of the Babylonian exiles; and his college was in Tiberias.
An objection was raised: There is a Boraitha: "Job's age was from the time when Israel came to Egypt until he left it." Read: "As many years as the Israelites were in Egypt." Another objection was raised. There were seven prophets who have prophesied to the nations, viz.: Bil'am and his father, Job, Eliphaz the Themanite, Bildad the Shuchite, Zophar the Na'amathite, and Elihu ben Barachel the Buzite. (Hence we see that Job was a Gentile?) And according to your theory, was then Elihu, just mentioned, a Gentile? He was certainly an Israelite, as it is written, "of the family of Ram." And why is he called a prophet of the nations? Because his prophecies were for the nations. The same can be said concerning Job. But did not the Jewish prophets also prophesy for the nations? The Jewish prophets prophesied to Israel, and to the nations also, but the above-mentioned seven have prophesied for the nations only.
There is an objection from the following: A pious man was among the nations, and Job was his name; and he came to this world only for the purpose of receiving his reward. The Holy One, blessed be He, however, brought chastisements upon him, and he began to blaspheme; the Lord then doubled his reward in this world, so that he should have no share in the world to come. (Hence we see that Job was a Gentile?) On this point Tanaim of the following Boraitha differ: R. Elazar said: Job was in the time of the judges; as it is written [Job, xxvii. 12]: ". . . deal in such vanities?" And which generation was one entirely of vanities? It is the generation of the Judges. R. Joshua b. Karha said: Job was in the time of Ahasuerus; as it is written [Job, xlii. 15]: "And there were not found such handsome women as the daughters of Job," etc. And in which generation were handsome women searched for, if not in the generation of Ahasuerus? [But perhaps it was in the time of David, when handsome women were also searched for [I Kings, i. 3]? There they searched only among the daughters of Israel, but in the time of Ahasuerus it is written, "in all the land."] R. Nathan said that Job was in the time of the Queen of Sheba, as it is written [Job, i. 15]: "When the Sabeans made an incursion." [And R. Samuel b. Na'hmeni said in the name of R. Jonathan: He who translates Malchas Sheba "the queen of Sheba" is in error, as the right translation is "the government of Sheba."] And the sages said: He was in the time of the Chaldea, as it is written [ibid., ibid. 17]: "The Chaldeans posted themselves," etc. Still others said that Job was in the time of Jacob and has married Dinah, Jacob's daughter. (They infer it from an analogy of expression, Nebala.) And all the just mentioned sages hold that Job was an Israelite, except the last, who maintains that he was a Gentile. R. Johanan said: It is written [Ruth, i. 1]: "And it came to pass in the days when the judges judged," etc. It means it was a generation that judged the judges. If, e.g., the judge said to them: "Takeout the toothpick from thy tooth," they answered: "If thou wilt take the beam out of thy eyes, I will remove the toothpick." If, e.g., the judge said to one: "Thy silver is become dross," the answer was: "Thy wine is drugged with water" [Is. i. 22] (i.e., if the judge accused one of a small transgression, the accused said to him: "Thou thyself art a greater sinner than I am").
It is written [Job, i. 6-9]: ". . . that the accuser (Satan) also came in the midst of them," etc. Satan said before the Lord: "I have sped all over the world, and found no trusty man like thy servant Abraham, to whom thou didst say [Gen. xiii. 17]: 'Arise, walk through the land in the length of it and in the breadth of it; for unto thee will I give it.' And not withstanding this, when he searched for a grave to bury his wife Sarah, and did not find one until he bought it for four hundred silver shekels, he did not murmur or bear in mind anything against thee." "Then said the Lord to Satan," etc. Said R. Johanan: That which was said about Job is more important than that which was said about Abraham, as regarding the latter it is written [ibid., xxii. 12]: "Now I know that thou fearest God," etc. And regarding the former it is written [Job, i. 1]: "And this man was perfect and upright, and fearing God and eschewing evil." What is meant by "eschewing evil"? Said R. Aba b. Samuel: Job was liberal with his money; it is customary, if a laborer has done some service to the value of half the smallest coin, that the employer takes him to the storekeeper, buys something for this coin, and gives the laborer the half due him. Job, however, gave him the whole coin for such services. "Then Satan answered, Is it for nought that Job feareth God? . . . the work of his hands hast thou blessed." What does this mean? Said R. Samuel b. R. Itz'hak: "Any one who took a coin from Job for business, has succeeded." And what means, "And his cattle are far spread out in the land"? Said R. Jose b. Hanina: His cattle have changed the order of the world. Usually wolves kill goats; Job's goats, however, killed wolves.
"But only stretch forth thy hand," etc. [ibid. 11-19]: "The oxen were ploughing, and the she asses were feeding beside them." How is this to be understood? Said R. Johanan: From this is to be inferred that the Holy One, blessed be He, gave Job a foretaste of the world to come (as about the world to come it is written [Jer. xxxi.] that pregnancy and birth in a woman occurred together). "A fire of God," etc. [to ii. 5]. Satan again answered the Lord, as said above.
"And Thou hast incited me against him," etc. Said R. Johanan: If this were not written it would be impossible for a human being to conceive it: the Scripture speaks of the Lord as if He were a human being who can be influenced through incitement.
There is a Boraitha: Satan descends and tempts human beings; then ascends and accuses them; then takes the order and takes the soul of him whom he has tempted.
"Then the accuser answered the Lord," etc. [ibid., ibid. 4-8], Said R. Itz'hak: Satan was more afflicted than Job himself. It is similar to a master who says to his servant: "Break the barrel, but save the wine" (without letting him have a vessel to save it in). So was it with Satan; the Lord told him to take Job's body, but to save his soul. Said Resh Lakish: From this we see that he who is called Satan is himself the evil spirit who tempts one to sin; and he himself is the Angel of Death, as he was told to save the life: from which it is to be seen that the life of man was in his hands.
R. Levi said: Satan and Peninnah both intended (with their accusation) to please heaven. Satan, who had seen that the Lord was favorable toward Job, feared that through the justice of Job Abraham's merits would be forgotten, and, therefore, he spoke as above. And Peninnah, as it is written [I Sam. i. 6]: "And her rival also provoked her continually, in order to make her fret," etc. It means for the purpose of making her pray and have a child. R. A'ha lectured the same in the city of Papuniah, and Satan came and kissed his feet for this.
"With all this, did not Job sin with his lips." Said Rabha: "With his lips he did not sin, but he sinned in his heart." What was it? [Job, ix. 24]: "Is a land given up to the wicked? He covereth the faces of its judges: if this be not the truth, who is it then? 1 Said Rabha. Job was about to turn the dish face downwards (i.e., to deny the might of the Lord). Said Abayi to him: Job spoke only about Satan. On this point Tanaim differ. About the just cited verse R. Elazar said: Job was about to turn the dish face downwards. And R. Joshua said to him: Job spoke only with regard to Satan. It is written [ibid. x. 7]: "Still it is within thy knowledge that I am not wicked, and there is none that can deliver me out of thy hand." Said Rabha: Job wanted to free the whole world of a trial. He said thus: Lord of the Universe, Thou hast created an ox with parted hoofs, and us without (and Thou hast commanded that only creatures with the parted hoofs shall be eaten, but Thou couldst have made it the reverse). Thou hast created Paradise, and Thou hast created Gehenna; Thou hast created the upright, and Thou hast created the wicked. Who can prevent Thee? (Hence no reward and no punishment should be dealt, as all was done according to Thy will!) And what have Job's colleagues answered to this? [ibid. xv. 4]: "Yea, thou truly makest void the fear (of God), and diminishest devotion before God." Which means that the Holy One, blessed be He, has created the evil spirit, and He has created wisdom as a remedy against him.
Rabha lectured: It is written [ibid. xxix. 13]: "The blessing of him that was ready to perish came upon me; and the heart of the widow I caused to sing for joy." From the first half of this verse we learn that be used to rob a field belonging to orphans, improved it, and returned it to them; and in the latter half we learn that if there was a widow whom no one wished to marry, he put his name upon her, saying that she was his relative, and then it was easy for her to marry. It is written [ibid. vi. 2]: "Oh, that my vexation could be truly weighed, and my calamity," etc. It was said by or to Rabh: The earth may cover Job's mouth for this. He makes himself a comrade of providence [ibid. ix. 33]: "There is no one who can decide between us, who could lay his hand upon us both." Said Rabha: For this also his mouth may be covered with earth: should a slave rebuke his master? [ibid. xxxi. 1]: "A covenant had I made with my eyes: how, then, should I fix my look on a virgin?" Said Rabha: He had not looked upon strange women, but Abraham had not looked even at his own wife; as it is written [Gen. xii. 11]: "Now I know that thou art a woman of handsome appearance," from which it is to be inferred that before that time he knew not that.
[Job, vii. 9]: "As the cloud vanisheth and passeth away: so will he that goeth down to the nether world not come up again." Said Rabha: From this we see that Job denied resurrection. [Ibid. ix. 17]: "He that bruiseth me with his tempest, and multiplieth my wounds without a cause." Said Rabba: Job has blasphemed by the tempest, and by the tempest he was answered. Blasphemed by the tempest--as he said: "Lord of the Universe! Perhaps a tempest passed before Thee, and changed to Thee the word Iyabh to Oyabh." 1 And by the tempest he was answered--as it is written [ibid. xxxviii. 1]: "Then did the Lord address Job out of the storm-wind. . . . Do but gird up like a mighty man thy loins: and I will ask thee, and do thou inform me."
So said He: "I have created many hairs on human beings, and for each hair I have created a separate hole; for if two should be nourished from one hole, it would blind the eyes of men; now from one hole to another it was not changed to me; and from Iyabh to Oyabh, should it be changed?" [Ibid., ibid. 25]: "Who hath divided off water-courses," etc. "There are many drops that I have created in the clouds, and for each drop there is a separate place; for if two drops should go into one, they would make the earth too soft, and it could not produce; these places were not changed to me." ". . . And a way for the lightning (that is followed by) thunders." "Many thunders have I created in the clouds, and for each thunder there is a separate track; for if two should go along the same track, they would destroy the world. The tracks were not changed to me; and from Iyabh to Oyabh, should it be?" [Ibid. xxxix. 1]: "Knowest thou the time when the chamois of the rock bring forth?" "The chamois of the rock is cruel towards its offspring, and when the time of bearing comes she ascends to the top of the mountain, so that the offspring should fall and die. And I send an eagle which receives it with its wings." "Markest thou when the hinds do calve?" "The hind has a narrow womb, and when the time of bearing comes, I procure a snake that bites her in the womb, so that she is able to bring forth the offspring. In both cases it must happen at the exact moment; for if it occurs a second before or a second later, the young in the first case, and the mother in the latter, would die. Now, from one second to the other there never is a change; and from Iyabh to Oyabh, should it be changed?" [Ibid. xxxiv. 35]: "Job hath not spoken with knowledge, and his words are without intelligence." Said Rabha: From this it may be deduced that one is not to be made responsible for his words at a time when he is afflicted. [Ibid. ii. 11-13]: "When, now, the three friends of Job . . . and they met together," etc. What is meant by "they met together"? Said R. Jehudah in the name of Rabh: They all entered at one time the gate of the city where Job lived; although a Boraitha states that each of them lived three hundred parsas away from the others. But who informed them? According to some, each of them had a crown on which were engraved the pictures of his three colleagues; and if one of them became afflicted, the picture was changed. And according to others, they had in their garden three trees, each of which bore the name of one of the friends; and if one became afflicted, the tree was changed. Said Rabha: This is what people say: "Either to have colleagues like Job's, or death."
It is written [Gen. vi. I]: "And it came to pass when men began to multiply . . . and daughters," etc. R. Johanan said: With a daughter, multiplication comes into the world, as in Chaldaic a girl is called rabhia; literally, multiply. Resh Lakish, however, maintains that with a daughter strife comes into the world, as rabhia means also strife. Said Resh Lakish to R. Johanan: According to your opinion, multiplication comes with daughters; why was not Job doubly rewarded with daughters, as he was with sons and with all his property? And he answered: Although they were not doubled in number, they were in beauty; as it is written [Job, xlii. 13-15]: "He had also fourteen sons and three daughters," etc. And farther on it is written: "And there were not found such handsome women," etc.