1. Any decent bike is a target for thieves. Lock it up.
2. Look on the side of the tire, it should say something like "26 x 2.15" or something. The second number is how wide they are. I have a set of Axiom Reflex Bladerunners on my MTB, they're plenty wide enough for 2" tires.
3. Only if you're planning on riding in snow.
4. I'd buy a new helmet if you're not comfortable with it. They're not that expensive, and hey, then you've got a spare helmet for emergencies.