Unfortunately I think it would be cheaper and more comfortable to just get an old beater car, like a mid 90’s honda civic. The money you would spend on gear, the bike, insurance etc… would end up being about the same, plus with the car you will be warmer.
Now that is out of the way, time for the motorcycle advice!
I am in the same position right now, I only have a motorcycle and I don’t have a car. One advantage that I have is I live in caliornia, so even the winters aren’t THAT bad, although they are pretty bad on a motorcycle. I wear full gear all the time, that means leather motorcycle jacket, leather gloves, textile overpants, full face helmet, and motorcycle boots. Even with all of that stuff in the winter mornings it can get COLD.
Because of that I invested in a balaclava and a fleece turtleneck to wear under my jacket.
http://en.wikipedia.org/wiki/Balaclava
With all of that I am pretty warm on my 25 minute commute, although it rarely drops below 40 F in the winter. I think it would be pretty unbearable. It looks like North Carolina where your college is located gets down to about 25 degrees F in the winter, thats quite a bit colder. Plus if it is that cold there might be snow, and I personally hate riding even in the rain, much less in snowy icey conditions.
Once again, it can be done, but this is my second winter with no car and right now it is getting pretty old haha. In fact I am saving up for a car right now, so hopefully next winter I won’t have to ride when it is pouring rain and freezing out.
Oh yeah one more thing, if you are going to be commuting more than 10 minutes I would suggest getting either saddle bags or this: https://www.bestbeginnermotorcycles.com/tourmaster-cortech-sport-tail-bag-review
It is no fun carrying a heavy load of books and stuff on your back, plus it could be dangerous if you crash. With a tail bag or saddlebags you can load them down and they will be safe and secure. I have even gone a further step and gotten a Givi tail case for my bike, its lockable, waterproof, big, and ugly. Perfect for the type of urban commuting I do.
I would also recommend you get the GS500, Ninja 500, or Vulcan 500 if you are going to be doing longer rides back home. You can make it on a 250, but I think the experience will be much more enjoyable on something with a little more pep. The gs500 is notoriously cold hearted, I should know it was my first bike. That means that in the winter you should store it in a garage where it is somewhat warmer than the outdoors, or perhaps hook it up to a trickle charger at night if you have a lot of problems starting it. I used to have to give the bike full choke and even then it would take me 2-5 tries to get the bike started in the winter.
Anyway, I hope that all helps and I haven’t deterred you too much, if you have some more questions feel free to ask. Riding a motorcycle is really fun, but I might be a little bitter about riding in all this cold weather! I’m glad it’s finally starting to warm up in california now 
It makes riding much more enjoyable.
Ben
~Best Beginner Motorcycles Admin
