Yeah, this game actually does have lots and lots of rules. They adjust them every year, too, as problems arise. I'll extrapolate a little on Sander's very good responses to your questions.
The player with the ball is "down" if any part of his body touches the ground other than his hands or feet and a player from the other team has touched him. So if the guy with the ball stumbles and falls down, but nobody touches him, he can still get up and run. If a player from the other team shoves him and he falls over, he's "down" as soon as his knee or elbow or butt touches the ground. Note that this is NFL pro football only: in college football the player doesn't have to be touched by the other team to be down, so if he stumbles and falls on his own he's "down."
It's perfectly acceptable for a player from one team to simply rip the ball away from a player on the other team. In fact, it's encouraged.
When one team is kicking or punting to the other team, the player catching the kick can "signal for a fair catch" by waving one arm in the air. A "fair catch" means the play is over once he catches the ball, and he can't run with it. The tradeoff is that the other team has to let him catch it without interfering; in fact he gets an invisible zone around him that the other team isn't allowed to enter until after he makes the catch. If he doesn't signal for a fair catch he can run with the ball afterward, but the other team can also hit him at any time, even in the middle of trying to catch it.
Punts and kick-offs are different in a few ways, besides just the act of kicking itself as Sander described. On either a punt or a kick-off, the result is a "touchback" if the ball is downed in the end zone. That includes the ball flying so far it passes through the goal posts. The result is the same as if the receiver had caught the kick in the end zone and knelt down: a touchback. The team receiving the kick then gets possession of the ball on the 20 yard line.
On a kick-off, the kicking team is not allowed to kick the ball out-of-bounds along the sides of the field. That's a penalty, and the other team gets the ball on the 40 yard line instead of the 20. On a punt, however, kicking the ball out-of-bounds along the sides of the field can be a good thing because it's not a penalty and the ball is considered "down" at the spot it left the field. In fact, it's sort of an art, called corner-kicking. If the punter for the team is so good at controlling where the ball goes he can make it fly out-of-bounds on the 5 yard line, that's where the other team gets the ball, and they never even get a chance to run it back. Another tactic punters often use is to try and make the ball flip in such a way that it hits the ground close to the end zone, but bouces straight up in the air instead of forward into the end zone (which would be a touchback).
Both the kicker and the punter are actually really important players on the team. In fact, the kickers often score the most points of anyone through the course of a season, and they have to have nerves of steel because the outcome of games can be decided by field goals at the end. That said, everyone makes fun of them just because they're usually the smallest, least athletic guys on the team, and the good ones often play until they're in their mid-40s, or even older. It's a pretty great job if you're good at it.