Is it legal to drop/shoot paintballs from an airplane?
Greg covered this very well in his answer.
The "miss by a surprising distance" bit is especially important here in the context of FAR 91.15 -- you would have to have a VERY large-scale paintball match for this to be viable, and some specific rules of engagement for the aircraft (always firing "into" the arena area as you don't want to miss and send a paintball into someone's car or kitchen window as that would get some distinctly unfriendly attention from the FAA, and probably the local constabulary).  

Operationally, if we're doing this with an airplane high-wing aircraft would obviously be the order of the day (Cessna 152), with a pilot (whose sole job is to fly the airplane) and a gunner/bombardier (who is responsible for hitting targets on the ground). You really don't want the pilot distracted by the game.
Your weapons guy is going to have to be able to account for the aircraft's speed and altitude in targeting (which is pretty hard with an adjusted scope/bomb sight - doing it without one is going to be tough).

As for shooting back at the plane, if we're talking about a "regular" airplane (something manned, like a Cessna 152) I would advise against allowing players to target the plane. At safe operating altitudes they're unlikely to hit the plane, but if they do aircraft skin is very thin &amp; dent repair is very expensive: If someone is lucky enough to score a hit it could be a costly "death".



Now, as a paintball player myself I will tell you that I find the idea of death from above when I can't shoot back at the aircraft to be rather unsporting!

What I would suggest instead is to deploy quadcopter drones. These can be had pretty cheaply and with some basic electronics skills they can be modified to carry/drop paint grenades (think Squadbusters) on your enemy.
If you want to get really fancy add a GoPro or some other remote camera to help with targeting, or add a paintball gun to the thing like these folks did.

The FAA's rules for "unmanned aerial systems" are still being written, but if you keep to the general rules for model aircraft (below 400 feet, and not charging anyone money for the drone's services) I think you'll be OK (when the final rules come out you'll have to review your drone for compliance).
As a bonus if the drone isn't too large and you have an indoor arena with liberally-minded management you may even be able to use your drone during indoor games.

(Kill rules for the drone and its operator are left as an exercise for the players.)