Chainer's last ability triggers if the entering creature was cast from a zone other than your hand, or if it wasn't cast at all and is entering the battlefield from anywhere (including from your hand).

If you somehow activate Chainer's first ability more than once during a turn (perhaps because Chainer left the battlefield and returned), you can cast a creature card for each ability's permission. Similarly, if another effect allows you to cast a creature card from your graveyard, you may use that permission and later use Chainer's permission to cast another creature card.

You don't choose a creature card to cast while activating or resolving Chainer's first ability. The ability creates a permission for you to cast a creature card from your graveyard later in the turn.

Chainer's last ability triggers if Chainer enters the battlefield without being cast from your hand.

Chainer's first ability doesn't change when you can cast the creature card. You can normally only cast it during your turn while the stack is empty, but if the creature card has flash, you can cast it at other times.