To OP -

It is not clear to me if your two flights are on one ticket or two different tickets.
Anyway, have them all available at Taipei airport, when you go to Thai Air Asia check in counter. [Added Later: There was some great advice here for most connections at most airports, but as dsquared37 points out below, it does not apply to OP's situation.]

There is no reason to worry about visa in Bangkok. Out of general prudence, I always carry my entire itinerary, a few pp size photographs and some small cash in US$, even if I don't expect to need them, and would recommend the same to you. In practice, all the laws, treaties, and Geneva conventions of the entire world carry less weight than the whim/misunderstanding of the guy/gal who happens to be on duty.

Good luck, and welcome to FT!