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.

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

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.