As already said, you do not need additional licenses. The usable features depend on the user's assigned licenses for sure (eg you can only use the Webex Calling API if you have to have a Calling license assigned).
A special API user is mostly not needed (they've been often used before service apps were available). Depending on you're use case, a bot or service application can be used. As you said you want to integrate it in your web application, Widgets etc might also be sufficient.
