For riding 6 miles your current bike is fine. No need to spend money on a new bike right now. Use the money to equip your current bike with puncture-resistant tires, fenders and lights. Get a rack and waterproof pannier(s). Since school means books, backpack may not be a good idea, rack and panniers will make your commute much easier.

No, don't get a fixie, they're just a fad, not practical, although many people will swear by how superior they are to regular bikes. Singlespeed but freewheel has its place but only if your ride isn't hilly.

But again, your bike will do just fine. You're already used to it. Ride it.

Adam