If a card with bank doesn’t resolve for any reason (such as being countered, or all of its targets becoming illegal if it has any), it won’t be banked.

Once you cast a banked card from exile, it behaves like any other spell. It may be countered, and it’ll be put onto the battlefield if it resolves as a permanent spell, or into its owner’s graveyard if it’s not a permanent spell.

If you cast a spell you don’t own, or gain control of a spell, you didn’t cast it from your hand, so you don’t bank it.

Banked cards begin the next games in a match in exile. They’re not outside the game.