Is a deleted API id recoverable?

Deploying the CF API caused the manually created one to be deleted, presumably because it shared the same name.

This shouldn't have been the case. Are you positive that someone did not manually delete the other API?

Can I "recover" the ID/URL of the deleted API? If it can be, I can manually create the resources needed.

Unfortunately this is not supported. Ids are assigned randomly at creation and cannot be overridden.

I am assuming that the use of a custom domain name for the API would be the correct solution. Yes?

Correct, this is the recommended method for maintaining a permanent URI for your API.

Are you positive that someone did not manually delete the other API?

Positive, yes.

The history of the original API is a bit convoluted, since it was subject to repeated development cycles. but it was not my intention to delete it. I performed a cloudformation package, created a changeset, and executed the changeset, resulting in a new API created, deletion of the old one.