Does imprison only work as long as the pokemon who uses it is still in battle?
Yup
I had Imprison and Perish Song active when I was fighting Arceus and when my Pokemon fainted he was able to use it again
any tips on catching azelf?
Use a pokemon with the damp ability.
no good ones that can survive future sight
Level them up to 120. Or maybe use a ghost.
If a pokemon with damp is in battle, and the other pokemon uses and fails explosion, does it still take away pp?
Yup.