Lamentations was written after the destruction of the city of Jerusalem and exile of its people, so would put it after 587 BC. Because it doesn’t say anything about being restored and the temple being rebuilt (which happened around 520) it was most likely written in those exile years. We can’t miss the writing style, this is poetry; in fact, it’s dirge poetry (funeral poetry), which shouldn’t be a surprise given a name like Lamentations. If you’re not used to reading poetry, I’d recommended reading it out loud, it’ll flow better and you’ll get more out of it. You can do it quietly if you wish, but if you just read it straight as if it is prose (story telling) you miss some of the lyricism in the art form. (If the Lord didn’t feel it necessary to take it in as poetry, the Holy Spirit would have inspired it differently. Engage in it correctly.)
In some way or another, each chapter is an acrostic of the Hebrew alphabet, with each line or set of lines starting with the next letter in the alphabet. That’s why they are all 22 lines (there are 22 letters in the Hebrew alphabet) except for chapter 3, which has 66 lines as there are 3 lines per alphabet letter). It’s a bit of a high art form, this acrostic funeral poetry, but it’s certainly intentional.
We don’t know for sure who the author is. It could be Jeremiah, because once you get tagged as the guy who weeps and laments, they are going to start framing you for other lamentations that are out there that are unsolved. Or, could be someone else or a group of people. Not sure it particularly matters, the book itself doesn’t seem to care so perhaps we shouldn’t either.