1. Do tokens (Valor, Infection, Prophecy) stay on the character sheet if he is knocked out?

2. If yes for the first question, can a knocked out character discard a Prophecy token to stand up?

1) Yes, unless the rules for the tokens in the expansion rules say they don't.

2) No. The KO hero can only use skills and abioities explicitly permitted while knocked out. For most heroes, that's really just "perform a stand up action" and the passive abilities that trigger automatically.