How does priority work in these cases?
After any stack-using effect(spell, ability, or otherwise) resolves, or whenever any player would get priority, the Active Player, (the player who's turn it is) gets priority first, and then the NonActive Player gets priority after the active player passes priority.

A player may put any number of spells onto the stack before they pass priority.  

If multiple sources controlled by both players would trigger an effect at the same time, they're put on the stack in APNAP order. The Active player puts effects from his sources on the stack first, and he chooses the order for his own effects.  After all of the active player's effects have gone onto the stack, than the Non-active player can put his effects onto the stack, and he can chose the order in which they go onto the stack.  

This results in the Non-active player's spells resolving first.