Your homepage URL can simply be your game’s Itch page. Itch’s documentation answers your other questions: https://itch.io/docs/api/oauth

While you can make your game server the callback URL, the server will have no way to attach/pair the access token to the client that requested it. Because of this, it is more proper & secure to set the callback to 127.0.0.1 (localhost/loopback) at some super high port that’s unlikely to be reserved. This means the client will have to temporarily host an HTTP server, too, with which it can extract the token, and then send it to the game server from its own address.

That being said, you don’t have to set this up during development. Separate the game logic from the authentication, then add logging in through Itch at a later stage. This will really ease things.