Should I play a RAPID GAME while I study?

I mean I only offer 3 hrs a day on chess and I bound to see this guy taking 30 minutes per move wasting my valuable studying time..

who claimed multitasking was ever a virtue?

I think it makes sense to play games, then analyze them, and try to tie your study in to problem areas you find through the analysis of your games.  If you can apply what you learn in actual games, you'll remember it better.  Practice is important, because it gets you used to playing, gives you experience, and develops your ability to think on your feet.

Adolf Anderssen won the first international tournament in 1851, but then after years of inactivity lost a match to Paul Morphy in 1858.  Anderssen noted that it's impossible to keep one's skill locked away in a glass case, but suggested that you had to develop it through games against strong opponents.