I've heard there is a chess.com group for players who have played over 1000 games here. Well, I've had a good look and I can't find it. Does it exist?

I dont know, but there is a tournament coming up for players over 2000 games, you just need 7 more... : )

http://www.chess.com/tournament/2000-games-completed-tourney

There are several groups based around 1000 rating points. Is it possible someone mentioned those?

Does this include only turn-based games, or all games? Of course, the live chess records, ratings excepted, were erased a few months ago and we started over (so my 2000+ is unverifiable).

I think it is turn based only.

That's a lot of turn based games

Yeah, took me a full 9 months...

I've been playing turn-based chess since 2003, and have played over 100 games on four sites, and at least a dozen on probably ten more. I imagine my total to be a tad under 2000. I do not play less than three days per move except in very rare cases, as when I fail to notice the time control.