E. W. Bullinger is reported to have said that Revelation was written primarily for the Jews. Was the writer's Greek sending a message of some sort?
No, a halting use of Greek isn't much else than a halting use of Greek. I also doubt that the audience of the book (in its present form) is primarily Jewish or Jewish-Christian. The internal evidence shows that the writer was primarily concerned about Christians in Asia Minor who were in danger of practicing idolatry (specifically the imperial cult and its patron goddess Roma, parodied in ch. 17). That suggests a special concern for Gentile Christians who came from pagan backgrounds, and the form of expression in 2:9 and 3:9 (with parallels in Ignatius of Antioch, who also wrote to the churches in Asia Minor) is also suggestive of a primarily Gentile community.
The writer confesses that he was extremely agitated by the visions he was receiving; would that not exacerbate the problem of accurately expressing his thoughts?
Actually, that is an apocalyptic cliche; the motif of agitation is found throughout the gamut of apocalyptic literature. Here the author is influenced by Daniel 7:15, 28, 10:7-19 (compare Revelation 1:17). It is also worth noting that Revelation is the most intertextual book in the NT; the author may well have experienced visions but the literary form of the vision reports in Revelation show a very deliberate and skillful reappropriation and weaving of traditional OT material, while respecting established apocalyptic conventions. These are not spontaneous ravings of a visionary but show a complex process of composition.
Also, could the Gospel of John have been dictated by John to a person more well-versed in Greek?
Anything is possible, but what I wrote above applies just as well to this scenario. It isn't a simple matter of superficially improving the Greek. It extends to the very level of composition and the ideas contained therein. This unknown person "more well-versed in Greek" would still be mostly responsible for composing and writing the book, even if inspired by the teaching of a "John". That may well be the case. Or a "John" wrote it who was different from the "John" who wrote Revelation. The question I raised in the previous thread is: which John? It is noteworthy also that the book technically is anonymous and only refers obliquely to its touchstone (in ch. 21) as the "beloved disciple," whose identity is entirely open to debate. The title of the book and its traditional ascription date to the late second century AD, at which time John the elder and John the apostle were already becoming confused with each other.