If I use Birthing Pod to sacrifice a Glen Elendra Archmage, does her Persist trigger bring her back before or after Birthing Pod's ability resolves?

Interesting question. What I think would happen is that the persist triggers immediately when the Archmage dies, before the Pod's ability goes on the stack, meaning the Pod's ability resolves first, then the Persist ability, as per "last in, first out." I may be mistaken on that though...it really depends when exactly the Persist ability goes on the stack.

Sac'ing the creature is an activation cost, so you'll pay that first. Since the creature dies, it's persist will happen, and then the Pod's ability will go on the stack. So it goes like this:

Tap Mana
Tap Pod
Sac Creature
Creature returns with persist
Search your library
New creature comes into play
Shuffle library.

I could be wrong, but I am almost positive this is correct.

Seems right, I've seen it done that way with Undying anyway.