Can you have a paywall on a mumble server so users will have to pay to use the server? Is the following functionality possible for a Mumble server to automate access tokens for registered users?

1. An API call to see if a register username is taken on the server
2. An API call to give a registered user an access token
3. An API call to remove a registered users access token
4. Disable voice and text chat on root room on a server
5. Kick users that stay in root room on server after X seconds

And is it possible to give an access token to a user who has not even connected to the server?

For one to three, I would implement that through an authenticator that replaces or integrates with Mumble authentication. See mumble-scripts/Authenticators for examples.

For four, I would declare groups in Mumble ACLs and assign the appropriate groups to the users to control permissions.

For five, I would create another Ice script which handles the custom timeouts according to user group/state and time.

I'm not very familiar with the newer plugin system and whether that would allow better integrated and I don't know what you are looking for in that regard, given that you specifically asked about making API calls, presumably from inside Mumble. The classic Ice API does allow extending the user context menus with custom functionality and you could respond with information to the user.