* Is this even a good idea? We're an all-developer team and don't have much experience in the space of working directly with designers. - An intern is inherently someone who is learning, who needs coaching. If you have no experienced designer for them to work under, they're likely going to have a hard time and you're going to have a hard time working with them. If the intern doesn't have experience working with developers, and you don't have experience working with designers, it could be a recipe for disaster. If, on the other hand, you hired someone who has experience working with developers, they may be able to walk you through it.

* Is it reasonable to expect him/her to be able to provide much help with our overall application look-and-feel, or does that tend to require a more highly specialized designer? - You definitely want someone with UI design experience... a "designer" who's just made a couple of logos and a flyer for their university club isn't going to cut it.

If this is an afterthought, hire an intern and see if it works out. Look at their work samples and ask - if I got this same quality of work from them, would I be happy with it (aka don't expect anything better than what's in their portfolio).

If this is essential to your application and delivering value (I feel UI/UX often is) then you should find the resources for an experienced professional instead, even part time, instead of a full-time intern.