@ZhenLin: As I mentioned, I'm no longer expecting 1 to be true, as I've attacked it in multiple ways and nothing seems to be fruitful, and I know that we shouldn't expect epimorphisms in $Lex$ to be epimorphisms in $Fun$ in general. I thought I'd pose it along with the others just in case there's something I'm overlooking (just now I realize I forgot to mention that $\mathcal B$ is small as well, although I don't think that will make too much of a difference). Out of curiosity, what is the counterexample in the case you present?

@QiaochuYuan: I mean "I'd like this to be true." I'm working on extending Deitmar's results, so there's not a guarantee that these claims should be true (although I do have reason to believe that they should be). However, that's why I'm also interested in question 3; if the results aren't true with the assumptions already made, I am not opposed to adding some other (reasonable) assumptions about $\mathcal B$ to salvage the statement.