I think the rules are unclear. But I imagine most GMs would allow it, especially on death saves. I'd be pretty peeved at a GM who didn't allow it on a death dave
If the animals have a Hero Point, sure. :)
Or as much as NPCs can, so I'd say no, not by the standard rules.
Being an aspect of the PC's power level makes this tough and I might override that rule unless foolishness was the major cause of the risk. Then the AC's death becomes a hard-knock lesson.
You can spend your Hero Points in one of two ways. Neither of these is an action, and you can spend Hero Points even if you aren't able to act. You can spend a Hero Point on behalf of your familiar or animal companion.