They do. Something to keep in mind is that just because a game has anti-cheat, doesn’t mean it won’t run on Linux. There’s a useful website that tracks games with anti-cheat, and whether they work on Linux or not.
Not really, there are several mod lists with varying levels of changes, that can be installed automatically, and most mods otherwise work.
I’ve been playing since the 12th, it hasn’t gotten better since then and my friend who got it day one hasn’t said anything about updates improving the performance.