I am very, very novice at homology theory so take what I am about to say as probably wrong (and possibly gobbeldy-gook). That said, since no one has answered your question I'll take a stab at my interpretation (see below) of it.

So, what is the in ? Is it the cellular decomposition (I have Hatcher, but not with me. And, I haven't read it yet)? If so, have you discussed the fact yet that the homology is independent of the decomposition? If you have then I think I can help with a) (maybe). Why can't you decompose into a zero-cell, two one-cells, and one two-cell in the natural way so that our chain complex becomes . I think (giving the outward normal orientation of course) it's easy to show that and are all trivial. From there it's pretty easy to calculate that