Tigerclaw

Does Tigerclaw's poison marker go away at the end of the round?

Yes. See page 8 (end of round), not the markers section on page 18.

Is leveling up done in secret, or should the table know which card has been added and removed from each deck? For example, can a Tigerclaw player know her opponents' attack types and whether they changed them this level up, or is it all hidden in the moment?

Follow the rulebook. It specifies the exact order for leveling up, and what constitutes hidden information. You need pages 4, 15 and 16. TLDR: you cannot see what others picked when you are picking upgrades.

Do tokens like Tigerclaw's poison or Bain's marker get removed at the end of the turn?

Check Page 8, paragraph 3 of the rulebook.