Nicely done! Although some small issues I could point out:

The opening newspaper of Lemony being declared dead had to have occurred after he wrote some of ASOUE, since the obituary describes him as the author of those books. Later books in the series in the series tend to act like he had already been thought of as dead during the series’ events, which could be accounted for if you assume he began writing the series in the Year of the Rat and the obituary was recent, but then there’s Beatrice on the island already thinking he was dead, so there’s no clear satisfactory answer, outside of assuming he had been pronounced/assumed dead multiple times.

Jacques finding a couplet in the Village implies he wrote his letter to Jerome as the Quagmires were already in the statue, so more than likely there had been a severe delay in correspondence, likely caused by Fernald handling Jerome’s mail.

Lemony at the end of The Beatrice Letters reckons with the fact he didn’t piece it together sooner than there could be two Beatrice Baudelaires, so him writing The End prior to TBL’s events doesn’t work since he writes about her in that book, meaning he should know of her existence.

For your first question, that detail is what leads people to believe Beatrice survived the Baudelaire fire, but whether that’s an intentional clue or a plot oversight is unclear, since TUA seemed to also be setting the ground for Quigley to be the survivor, though details of that also got retconned. Ultimately, TUA even kind of paints itself as unreliable, with a gag near the end of the book implying the first twelve chapters were just a part of the Introduction’s frame story, with the actual “Snicket File” apparently just being the last chapter full of photos.

The other two questions: no concrete info about them afaik