You must choose all the targets at the time you cast Put Away. You can’t use it to counter your own spell and then shuffle that card into your library because at the time you choose targets, the spell isn’t in your graveyard yet.

If the targeted spell can’t be countered, that spell will remain on the stack. Put Away will continue to resolve. If you targeted a card in your graveyard, you still get to shuffle it into your library.

If you choose to target a card in your graveyard, Put Away will have two targets. If either one of the targets becomes illegal, Put Away will continue to resolve and affect the other target.