1st- What lenses are compatible, any m42 or will shorter focal lengths cause vignetting?
Any M42 mount lens should be compatible. The Pentax lenses are particularly good for this camera. Almost M42 mount lenses (other than the Meteor lens that comes with it) are built for 35mm still cameras so no vignetting; even in Super 16.
2nd- What film works best with the k-3 unmodified? I want something balanced for tungsten but can be shot in daylight with a filter.
Any and all 16mm camera films will work just fine. Kodak Vision 3 200T would make a great choice, so would the 500T. I love 50D outside; it's the lowest grained film Kodak makes for cameras these days.
3rd- Is it worth it to get the light meter battery adapter, or should i use an external meter?
I've always used an external meter. Much more accurate, and even if you get the internal meter working it's probably old and not reliable.
4th- Is the super 16 mod worth it?
Depends on how much you spend. I actually noticed a difference when mine went to Super 16 with the same film and shooting conditions. Widening the gate is easy, the trick is to find a re-tooled mount to re-center the lens. Not absolutely needed but helpful if you're using the Meteor zoom. You'll see vignetting with the stock lens in Super 16 without re-centering the mount... and probably a little at the widest setting even after that.
Another easy mod would be to go Ultra-16 and just widen both sides of the gate. Plus is that it's cheap to do, downside is that not every telecine house can handle images between the sprockets.