Getting credentials from mysql db
they would somehow only needed root credentials to somehow get user
  and password from "mysql db" and then use them to create the
  connection using customer db credentials. Is this possible?


No, this is not possible, passwords are stored hashed. Hashes are one-way functions where if you get the password, you can get the hash, but it is virtually impossible to do the opposite.

Your API must store temporarily the password, somewhere when it is created (probably on memory), and then send it securely to the client application, then discarded. If that is not desired, you should try a different authentication method that doesn't require passwords or shared secrets.

MySQL authentication should be separate from an application authentication. You could store user credentials (properly hashed and salted) in a database table, but you should keep the users of your application and the users of a database separated (or use an external authentication mechanism, as suggested above).