I took my laptop into futureshop about a month ago as the screen was not working and also not working on an external screen. They told me it was the motherboard and changed it, I took it home and it worked just fine. It wasn't until I had used the laptop not plugged in and then plugged it back in that I noticed it wasn't charging and it showed "plugged in, not charging". I took it back to future shop 2 weeks ago and I finally got a call saying it was back, however it was an issue with the battery and I am liable.

I am wondering if it's possible...
a) its not the battery and the new motherboard affecting it (I've read this can often be an issue)
b) somehow while future shop had it in their possession the battery got screwed up OR
c) it was installed incorrectly and somehow killed it

I find it highly unlikely that hours after returning home it just HAPPENED to break!! I'm frustrated because I'm a broke student and it has a great battery that lasts through hours of lectures... I need the battery and don't really have the cash!!!

Yes. The charging circuit may have quit working on the system board. They may have given you the wrong power supply, although you probably would have noticed. Anyhow, make make sure it is the same AC adapter you left with them or at the very least the correct voltage & amps. They could have even given you the wrong battery (That probably wouldn't be an accident, though)
Whether futureshop is to blame or not, who knows, but batteries do die and will do so abruptly sometimes. It could be a coincidence and most likely is, if the battery is indeed bad. It's probably feasible that a used battery left sitting in front of a sunny shop window for a few days might take a dive.
Nope, can't really think of a way that could happen. Batteries will only fit in one way and there's really nothing you can do to the system board that would cause an issue, unless they managed to short out something with a loose screw etc (improbable).