As far as your skin condition, there are a lot of variables that you might not be aware of yet. Navy ships use a lot of "stuff" that you won't find out in the civilian world. Between paints, lubricants, materials, etc; there are a lot of things that could make your condition worse.
Case in point, I had a student one time that made it all the way through Prototype just fine, and went to my old ship. When he got there, he started having bad reactions with his skin. Turns out, he was allergic to something on the ship(don't know if they ever figured out what exactly). So he ended up not being able to serve on board the ship because they couldn't control it. I believe that he was discharged after that.
Another case involves an MM who was allergic to the oil we used in machinery. His skin would crack, peel, and blister sometimes when he would handle the oil. He was able to control it using gloves when he needed to contact the oil.
So it all depends on if your condition is controlable in a reasonable fashion. However, be aware that it could become significantly worse when exposed to elements that are associated with Naval vessels.
You stated that you were "tired" after going to several different colleges and universities. College is a joke compared to nuke school(at least that is my opinion about college after surviving nuke school). Be advised that you can't "transfer" easily in nuke school just because you are "tired". Furthermore, a 2.0+ gpa isn't exactly steller, and based on that, you are going to be hard pressed in school. Your study hours are inversely proportional to your GPA. Based on the GPA you provided, if this was nuke school, you would be in class from about 0600 to 2300(6 a.m. to 11 p.m. for you non-military types) every day of the week and putting in almost 8 hours a day during the weekends. Of course, I could be wrong......