I created a gift card order from the admin. Who receives the gift card code?

At the moment, no one. Create order saves a gift card product as an ordinary line item, so no gift card code is created, whether you choose Paid (Processing) or COD (Pending). Moving the order to Processing later doesn't create one either.

To give your client their cards, issue them yourself:

1. Open Plugins → Gift Cards.
2. Under Issue a gift card, enter the Amount.
3. Enter the Recipient email of the person who should get the code. The code is emailed there when you issue it. Leave it blank if you'd rather share the code yourself.
4. Optionally fill in Recipient name, Sender name, Message and Expires on.
5. Click Issue gift card.

Repeat for each card. Every code you issue is listed under Issued cards.

Keep the order you created as the record of the client's payment. The cards you issue from the plugin aren't linked to it.