I noticed you unlock weapons/items every time you level up, but is that all there is to it? Do the characters stats increase at all?
No, the character stats do not increase.
I wish they implement some kind of handicap for servers also, so beginner players can play with beginners and experts with experts, mainly to avoid experts kicking beginners from servers if they don't play as well as they do (like it happens now on L4D2 for example).
There is a handicap. It's called "Easy, Normal, Hard, Insane." Yes, we do kick level 1-9s out of Insane groups unless they have a prestige star and rightly so. It is true that you can complete Insane missions at those levels but the fact that they aren't weapon leveled and don't know the map layout is a very valid reason to boot someone.
Easy-Hard, sure but why would someone just download a game and pop into a hard server their very first run through? You get maybe 50-100+ xp extra for higher difficulties, so its not like you can hump people through levels.
Learn the map and don't shoot your teammates. You'll most likely be welcome in any group.