The game and matchmaking on the 360 operated better before Christmas. Since Christmas, Live has been experiencing issues and much of your frustration comes from that.
Games have been taking longer to find and to fill up, there's been an increase in lag and parties have been dropped or unable to connect.
Losing connection when the host quits is a CoD problem, but it is being addressed, as the next patch will include host migration.
The player limit feels fine to me, except in Tactical perhaps (3 vs 3). More players would turn it into a wild melee and the current 6 vs 6 is well suited if you are looking to apply tactics and teamwork. You can play Ground War for an 8 vs 8 experience as well.
If you play the same game on two different systems, you are going to prefer one over the other and there's nothing wrong with that. If that's the case, simply stick with your preference.
I can guess what the lean left and right mean, but I haven't seen the feature so maybe you could explain it? If it allows you to move your upper body to the side and limit your exposure to fire, that might cause a balance issue using an analog controller that doesn't exist when you use a mouse. If we're talking about dodging around like a boxer, that just sounds like a comical thing to do in a gunfight.
Matchmaking instead of server selection will likely continue. I recall being able to select from a match list in PDZ and Gears of War. In both cases, it was abuseable for boosting, so I don't see any Live games offering that option in the near future. I would like to see it happen as well.
If you decide to keep it to play on Live with friends, it will require some adapting to, but I think it would be a comparable experience to playing on PC with people you don't know. I own Halo 3, but I don't like it, I only play it with friends and that is enough for me to enjoy it on occasion.
Best of luck whatever you decide to do.
