1. That is how I got my first Betta home from the pet store 3 hours away and she did just fine.
2. That is a good idea for adjusting him to the colder air, or you could put him in his cup right away, stick a small thermometer inside his cup or one of those sticky ones on the outside and put a hand warmer around his cup to keep him warm. Don't worry about the music, as long as it isn't blaringly loud, it shouldn't bother him, in fact, he may enjoy it. My babies enjoy music.
3. That is fine, but if you have a hand warmer around his cup, a quick stop for the restroom/snacks whatever, he will be ok.
4. If you are not using distilled water now, don't use it when you get home. A good quality water conditioner will neutralize the chlorine in the water in your house and it will be fine. If you are worried, have your mom fill up some jugs now, and let them age until you get there, then add your conditioner when you get there before you put it in your tank. You can also bring several jugs of your water from your college to fill your tank if you want to. Bettas can be very sensitive to changes in water parameters, especially from tap to bottled water, so you could send him into shock if you don't acclimate him slowly over several days to distilled water. Plus distilled water lacks the essential nutrients your fish needs to stay healthy.
5. Keep him in his cup with the hand warmer if you go that route until your tank has warmed up, then float him in his cup for a while to let the temperatures stabilize. If you are using your mom's water, you can acclimate him to that water by exchanging a spoonful of water from the tank to his cup every so often over several hours. If you bring water from your dorm, you can skip the water exchange part and just worry about the temperature acclimating. If you don't use a hand warm, just float him in his cup with both his tank and cup at room temperature, and wait until the tank is at the desired temperature.
Good Luck, he will probably be fine, just make sure you acclimate him properly and keep him warm - given his recent illness, it might be better to go with the hand warmer option around his cup so he doesn't get cold and compromise his immune system. Another option is to put him in a small cooler with a hand warmer so the heat stays in and place him on the floor of the passenger seat or secure it in the passenger seat. And most importantly, relax.
I hope this helps - if you have any questions, please ask.