Q u o t e:
If our pet dies,
Can we use Feign Death, drop combat, and instantly bring out one of our 5 different pets in that short window of being out of combat in an Arena Match?
No, because you have to dismiss a pet before you can summon a new one. You also need to rez a dead pet before you can dismiss it. It's possible that allowing all of that in combat is fine since it would take so long to actually pull off and in the meantime you could just have rezzed your pet (or healed an injured one rather than dismissing it).
On the other hand, dismissal no longer affects happiness in any way since the new model is you can swap them more regularly. Your pets just have to learn not to get all emo about it.