How do undying triggers resolve against exile and other players reanimating my graveyard?
Your first example works as you want: the stasis snare doesn't exile your creature. When it moves between zones, unless otherwise stated, it and everything that targets it forgets that it was something else. Note that 400.7 says zone, not just public zone. 

For your second example, you get to keep your creature. Grave Betrayal triggers at the beginning of the end step (technically, it creates a delayed trigger that triggers at the end step), while Undying triggers immediately. Once Undying resolves, it's back on the battlefield and is a different object than what Grave Betrayal is attempting to steal.