A 20ga or a 16ga is more than adequate. Rabbits are not particularly hard to take. Big fox squirrels are harder to deal with than a rabbit is.
The key is weight -vs- how far you have to travel while tracking the dogs. I started back out a few years ago with my 12ga Ithaca 37. After 1 day of that, I had enough and went on a search mission for a 16ga Ithaca, which I found in a pawn shop.
The older ones, ie: pre- serial nbr 855,000 are lighter. The downside is that they have barrels fitted to the action. (ie: Ithaca 37's).
One of the old Stevens 77 series guns (in a pump) in 20ga (I have one of these also) work quite well and aren't that heavy to carry.
If your budget allows, a Remington 1100 in 20ga would be a good rabbit gun.
The key is what will fit you and your circumstances. If you are hunting private grounds and don't have to hump over hill and dale, then your existing gun will do just fine.
Here are some articles that will help resolve what you are asking about whether the 12 is superior ballistic wise (ie: reach out there) which it isn't, except in some very specific situations where the hull capacity comes into play.
the first is a link to all the articles, spend some time reading. I pointed one out specifically, but I think you can find the answers to some of what you are asking and generate some new questions. use Google, it is your friend, just remember though, everyone has their opinion on the best gun, gauge, dog, woman, car truck etc
BUT
the scale doesn't lie, a 5lb gun is easier to carry than a 7 to 8 lb gun and shotgun ballistics are shotgun ballistics. you just have to do some home work.
http://www.chuckhawks.com/index2c.shotguns.htm
http://www.chuckhawks.com/twenty_vs_twelve.htm
http://www.fieldandstream.com/articles/guns/2002/03/wonderful-20-gauge