I never even realised it was a thing people did commonly until I started watching Critical Role.
The closest my group has ever come is playing the Hell's Rebels adventure path for Pathfinder. In that you resurrect an old group called the Silver Ravens, so that essentially becomes your party's name.
I haven't played Wasteland 2, but I quite enjoyed Wasteland 3. I've played it through a couple of times. Might be worth giving it a go?
Apparently some people don't like that you asked that question.
I honestly couldn't say. Usually I'd be able to give you a definite answer one way or another (and maybe my answer would be different if I'd played the game first) but I don't think there's a huge difference. Maybe just go with whatever you get a hold of first? That said, you can never go wrong with experiencing the source material first; at least not when the adaptation is of a similar quality. You'll have a good experience either way. It really just depends on whether you want to watch the show and say "oh that's how they did that in the show" or play the game second and say "Oh, that's what it was originally like!"
I'll preface this by saying that I haven't played it in probably at least two decades, but I used to love Wacky Wheels. I believe it's on Steam now, too.