First off, I'm Army, so I can't really speak for ay of the other
(lesser) branches.
1. I have some college education. Does this help me or hurt me in any way? I haven't yet completed my bachelor's, but should in a little over one year.
It will help. You can get up to E3 right off the bat assuming that you have enough credits, and E4 with a bachelors. My three semesters of college certainly haven't hurt me. It will put you a little above the 17-year-olds next to you.
2. I'm overweight. I don't want to be the fat guy that can't complete basic training. I intend to work on this before I ever enlist, but will weight be an issue in training? (I assume I'd lose some of it through training, anyhow.)
It depends on how much your weight hurts your physical performance. As long as you can meet the minimum standard, the DSs won't care. Once you get into the regular Army, PT scores are very important (of course different MOSs will be different). It will affect your promotions, what schools you can go to, etc. But it is always something that can be improved. Assuming that you meet the entry requirements for height/weight and push yourself hard, you should be fine.
3. My personality. I'm quiet, reserved, and introverted. I hate to sound like a wuss, but I've always been apprehensive about new experiences. It's a stupid question, but would I get ripped apart by my peers or commanding officers?
'Quiet, reserved, and introverted' describes me perfectly. You won't get ripped apart. Just don't hide in a hole or be a barracks rat, that's all. You don't have to be everybody's friend or the platoon daredevil to get along in the military.
4. I've never fired a gun in my entire life. Would this present a problem in training? (I assume you're taught what you need to know, but does a lack of prior experience present any problems?)
Prior experience is certainly good, but not necessary. The military caters it's training to the lowest common denominator. There were several guys in my basic platoon who had never fired a weapon before, and they all shot fine by the end.
To be honest, Basic (OSUT for me) is rather short in the long run. It's even kinda fun after you get over the culture shock and learn enough to know how to stay out of trouble. While there, your goal is to blend in. Never be first or last. That changes once you graduate, but overachievers just buy themselves unwanted attention in Basic.
Once you get to your unit, stay long enough that you are no longer considered a 'cherry', and prove yourself competent and intelligent, life gets a lot easier.