peacefulpete, in regards to 1 Esdras 4:45 an Oxford Study Bible edition of the NRSV refers to 1 Esdras 1:55 which is part of the passage about the Chaldeans (Babylonians) burning down the house of the Lord, with no mention in that verse of the Edomites being involved. However the note also refers to Psalms 137:7 which you quoted.and it refers to Obadiah 1:11-14.
Regarding Jeremiah 41:4 could that be referring to the burned down temple of YHWH (to its remains)? Note that Ezra (not Esdras) 2:68 - 3:6 (NRSV) mentions (in 2:68) those "who came to the house of the LORD" and note it uses that terminology, even though the temple was burned down and not yet rebuilt. 3:3 (NRSV) mentions some people set up an altar on the foundation (NSRV says "its foundation" but 1984 NWT says "its own site") in order to make burnt offerings on it, even though 3:6 (NRSV) says "But the foundation of the temple of the LORD was not yet laid." {184 NWThass similar wording.] I think the foundation mentioned in verse 3 is of the remains the burnt down altar site of the burnt down temple. I think that verse 6's mention of the not laid foundation is of the foundatio that would be laid in the rebuilding of the temple. This shows that according to the Bible even the ruins of the burnt down temple of YHWH was still called the house of YHWH and that some made burnt offerings there before the temple was rebuilt. What do you think about that?
As to who burned down the temple perhaps the Edomites did not themselves burn it down (despite what 1 Esdras says), but rather that the Edomites' demands caused the Babylonians to burn (and/or tear) it down. Thus, maybe the Edomites could be said of burning it down in the sense that was because of their urging that it be burnt down that the Babylonians decided to burn it down, but with the Babylonians being the ones who did the actual burning.
Similarly, according to the gospels the Jewish people and Jewish religious leaders urged the Romans to execute Jesus on a state/cross (and as a result many people throughout nearly 2000 years have said say the Jews killed Jesus), but those same accounts says the Romans did the actual execution of Jesus on the stake/cross. But the accounts also say the Pilate only gave the execution order because of the urging of the Jewish crowds.