Sleep or shut down?
When not in school my machine pretty much stays at my desk and i dont put much use to it besides video and audio playback. This being said it basically stays plugged into the adapter all day long, should i remove the battery in this case? Another question is, should i shut my computer down every night or put it to sleep?
no, you should not remove battery when plugged into AC
although it's safe to do so, the MB or MBP will automatically reduce the processor speed when running on AC without a battery - see this technical note from Apple. As the note says, this is in case the computer needs more power than can be provided by the AC adapter alone - as could possibly be the case if you're running a lot of video and the CPU gets heavily loaded.
as for sleep or shutdown - I run my MBP plugged in every day at work for 8hrs, then on battery at home, and always just sleep the machine. There's no need to shut down every day.
OTOH, I would recommend you periodically recalibrate your battery once in a while, to make sure you're getting the best performance out it if.