Hi there!  First of all, I admire your enthusiasm. ;)

I’ll try to answer your questions as best I can, and if anyone else has input feel free to chime in.

Rachel and Kurt are still NYADA students. Remember in 5x02, Kurt visited Rachel in the dance room as she was tuning a piano. She said that Carmen needed volunteers for the NAYDA annual piano tuning. There was also a number of students in the room, and Kurt had his canvas bag on his shoulder. Later, in 5x04, Kurt held Pamela Lansbury auditions in the dance room. I don’t think he would do that if he wasn’t a student. So far this season, there has been a noticeable shift in focus for NYC from season 4. For whatever reason, the writers have focused more on Kurt and Rachel’s activities outside of NYADA classroom  (e.g, the Spotlight Diner, Funny Girl rehearsals, and Pamela Lansbury). 

They aren’t on summer break yet. I know the whole timeline seems confusing because of this extended school year deal. Based on what’s happening in Lima with Nationals and graduation approaching, I’d put the NYC side right at late spring (April/early May).

As for the free time, that’s the decision of the Glee writers. We only get glimpses of Kurt and Rachel’s daily lives. College is also much different from high school in that the class schedule isn’t as regimented. There are usually pockets of free time here and there. It depends on the individual college student. This is also where fanfiction comes in to help fill in the blanks.  :) 

Hope that I’ve shed some light on your questions. Thanks for messaging me and reading my meta. If anyone has a different take or additional insight, feel free to add. :)