Yeah, that whole scene is just a hot mess. Columbia does have housing for students in its graduate/professional schools, but it's apartments in the neighborhood, not dorms. Not sure what percentage of the law students take advantage of that, but I'd expect the numbers to be much lower for locals.

Law students can petition to cross-register for classes at Columbia's other graduate schools. "Conversational language courses" are specifically excluded from being approved (seeing as they're not graduate-level courses in any event). I'm not sure where that leaves translation/reading language courses, but it's pretty obvious that Matt and Foggy are talking about conversational classes. The debate over "are you taking Spanish or Punjabi" is a very undergrad freshman interaction.
(http://web.law.columbia.edu/academic-rules/academic-procedures)

So.....it's a mess.