I just want to elaborate more on questions 3. and 4. I'll consider the locally isomorphic groups SU(1,1) of SL2(R) and SU(2) of SO(3)

There is an analogy between the discrete series of SU(1,1) and the unitary irreps of SO(3). Both have holomorphic representations on the group's orbit on the flag manifold S^2 = SL(2,C)/B (B is a Borel subgroup). In the case of SU(2), the orbit is the whole of SU(2) while for SU(1,1) its is a noncomapct supspace: The Poicare disc. In both cases the representation space is a reproducing kernel Hilbert space and the group action is throug a Mobius transformation. This analogy generalizes to other non-compact groups having a holomorphic discrete series and it can be considered as a generalization of the Borel-Weil construction for compact groups.

Concerning question 4. I think that you are talking about Wigner's theory of Lie group contraction, in which a Lie group with the same dimension and with more "flat" directions is associated to the original Lie group. For example there is a contraction of SU(2) to Eucledian group in two dimensions and SU(1,1) to the Poncare group in two dimensions. There are interseting connections to the group representations of the contracted versions, and also of the Casimirs.