It is hard to take college in the infantry. I had to get a college option as part of a reenlistment to take a semester. This may not be available in all areas. Correspondence and online classes are your friends here.
1. Yes. That doesn't me you will just because you ask.
2. see 1
3. There are Ranger and Special Forces enlistment options. They may not be available, you may not qualify for the option, and even if you get such an option, you may not pass the selection. I don't know if it is an enlistment option but you may ask about Sapper school.
If you get any of these schools in your initial contract you probably will only get 1.
4. No awards that I know of. The mountain tab is part of the 10th ID unit patch, just as Airborne tabs are part of the unit patch. It does not indicate any sort of qualification of the soldier wearing it.
I think there is an arctic school. Never heard of a dessert school. The Mountain Warfare school is run by a guard unit (Minnisotta I think?) and will be difficult to attend. I've heard of a jungle warfare school but I believe it's closed now.
5. see 3
6. Probably won't get sniper. Ranger, Airborne, Air Assult, SF, and Sapper could be available to you, but you will be a lower priority.
This is a ten level task