Routed Heroes cant - move, start a duel, attempt a quest and support a battle

I can't find anything that would prevent routed Hero to heal or train during his Quest phase. Is it allowed?

I assume that would be the same for Commanders.

Also -

Does hero counts as being in area while moving through it? There are some tactics cards that could be used while hero is moving through area.

(I guess not, since it could apply to units aswell and would mean that you have to kill them?)