Can you use the Punisher after the opponent has selected which stunned character he wants to recover to KO that character anyway? Or does the character immediately recover when selected?

No. Both players get priority at the beginning of the recovery phase, but after both players pass on an empty chain, the recovery phase continues and neither player will get priority again until the following draw phase. Since you won't get priority when characters are chosen for recovery, you cannot play Punisher's power.

Along those same lines is there a time between readying your characters and drawing that you can use longshot?

You can play his power in response to the "Each player draws 2 cards." effect being put on the chain during the draw phase, but that is the only time.