a) Weyl's unitary trick implies there are no nontrivial irreducible finite dimensional unitary representations of SL(2,R).  This is basically the opposite of SO(3).

b) [Wikipedia][1] has a classification of all unitary irreps.  Since you're working with functions instead of sections of a line bundle, you need to restrict to those invariant under SO(2), i.e., weight zero.

c) I think you get real-analytic Eisenstein series and discrete series.  Eisenstein series form a continuous spectrum, while discrete series give modular forms.  You can find more in Gelbart's book "Automorphic forms on adele groups"


  [1]: http://en.wikipedia.org/wiki/Representation_theory_of_SL2(R)