I graduated from MUSC, and it is an excellent hospital! (MS - Cytology and Biosciences).
Downtown - I assume you would want to be semi-close to MUSC. If so, Spring Street and Cannon Street have newly renovated condos and apartments on them. I did a little checking, and you can get a very nice place for low-end $1000 - $1500 high-end in that area. The high-end are brand new condos on Cannon Street, very nice. Having gone to school there, I think you would enjoy living close to MUSC much more than living in Mt. Pleasant! / especially if you will be working long or late hours. The bridge to Mt. Pleasant is very nice now, but if there is an accident on it or as you get into Mt. Pleasant, the traffic is terrible! The traffic in Mt. Pleasant can be terrible at anytime. Don't get me wrong, traffic downtown can be unpleasant too, but if you lived very close, it would not really be an issue. You would be around many more people similiar to you too downtown. Many people who live and work at MUSC live close, as well as many College of Charleston students. There are restaraunts, drug stores, shops, etc...in walking distance in the MUSC area. If you lived downtown, you could get a bike and get just about anywhere at anytime.
About MUSC--My son had heart surgery there, and he was admitted for Rotavirus for 8 days there too. I have had EXCEPTIONAL care and wonderful experiences there. The physicians and nurses are the best I have encountered anywhere. I also organize a toy drive every year with my Homeowners' Association for the Children's hospital. It is an amazing place for children, and a great place for learning as well. So, good luck with your decision.
p.s.-- Google Spring Str and Cannon Street properties and see what you can find. Bee Str., Rutledge and Ashley Ave. as well.
hope that helps