Not quite. Try using Panopticlick [0] for an overview of how your browser can be tracked. Less unique is better. There's also sometimes new methods discovered to track users, but that's not something you would usually care about.

Firefox has good settings you can set in about:config [1].

If you don't have WebRTC disabled, there's a good chance a service can get your public IP address directly.

[0] https://panopticlick.eff.org/ [1] https://www.privacytools.io/#about_config

Not entirely. Brower cookies, for example, persist outside any VPN session.

Make sure you're not using an operating system that would be unpopular among the IP block you're VPN-ing through. Also, better have the same browser, screen resolution, fonts installed, ...