If a hero is knocked out, does she refresh cards/equipment at the start of her turn as usual?
Yes, cards refresh while you're KO'd. By the same logic, a KO'd hero can also change equipment in Step 2, before the action phase begins. I'm not too keen on that myself, but there's nothing in the rules to stop it.
The only action a KOed character can do is stand up and then their turn ends. There has been a little bit of a heated discussion last night about standing up and then doing other things that don't require an action, such as drink an elixir token.
My understanding of the rules was they stand up, that's it, next player. That is the penalty for being KOed to begin with. Is this understanding wrong?
You are correct. Before the hero stands up they can't use any abilities, because they're KO'd. Their turn ends as a part of the Stand Up action, so they cannot do anything after standing up, action or otherwise. Their turn is already over.