Welcome to Cigars!

1. How important is it to keep your cigars in a humidor?
Very important for the long term for maintaining flavor. Important in the short term from burning properties. Storing cigars at the proper humidity maintains the oils and flavors as well as creating proper conditions for desired burn behavior. Allowing cigars to dry out will result in loss of flavor, hot burns ==> off flavors, and possibly cracking of the wrapper. A short term outside of the humidor for up to few days depending upon conditions prior to smoking is OK. Some people may leave a cigar in an un-humidified box as a practice of "dry-boxing" to slightly dry the cigar out before smoking.

2. Can you rehydrate dried up cigars?
Yes. But if they were dry too long, it is probably not worth it. They won't be the same. If it was not too long of a period you can, but should be done slowly and gradually, depending on how dry the cigar became and the length of time that is was dry.

3. How do you prevent "runners"? Where one side of the cigar burns faster than the other?
Often times you can't, if it is due to construction issue. The best you can do in this case is to touch up the slower burning side with a match or lighter.
Sometimes uneven burning can be caused by storage conditions, such as having a cigar stored too close to a humidification source on one side and thus one half of the cigar has slightly more moisture than the other.

4. Lacking a real humidor I've been keeping my cigars in a cigar box with a "water bag" (one of those bags with those gel things you let soak in water). I keep the whole thing inside a rather large Ammo can I picked up while deployed. This serves as my "travel case" as well since it pretty near indestructible. I keep a few pipes, some pipe tobacco, and a few other smoking tools in there as well. My question is this...how difficult would it be to convert this ammo can into an actual humidor? And is it ok to keep the pipe tobacco stored in there with the cigars?
Any container with a decent seal will work. Your set-up with a cigar-box in a ammo can works great! As someone already mentioned, they make Ammodors just for this purpose. As an alternative you can use a sealed plastic container.

I would not recommend storing your pipe tobacco with the cigars though. The flavors and aromas from the pipe tobacco with merge with your cigars. Being as you like the pipe tobacco, maybe this is a good thing for you! Where you have your cigars segregated in a cigar box, it might take some time, but will still occur. Infused and fire cured cigars should also be stored separately from other cigars.

5. Any of you guys/ladies in western New York? I'm pretty central to Buffal, Olean, and Geneseo.
I think there are a few guys up that way. There is a fine brother @WNYTONY in Olean and I believe he might occasionally enjoy the pipe as well.