I should point out that the crochet article linked above has references to research literature. In particular, it mentions that there are no C^2 embeddings of the whole plane, but there is a C^1 embedding.

When you say size at most Cn^e, do you mean that is the order of the group, or the order of its image in GL_n(C)?

You can formally exponentiate a Harish-Chandra pair. G_0 is a proalgebraic group, but G is a formal group with a big proalgebraic subgroup. You can think of it as a group ind-scheme.

I don't know if it is possible to analytically continue T to the lower half plane. Modular forms tend to yield essential singularities at the boundary of the q-disc - pick a Mobius transformation that takes infinity to a rational number, and feed infinity to a positive weight transformation formula. I'm afraid I also don't know what happens if you try to feed in irrational real numbers. I've heard people mention modular forms definable on the lower half plane, maybe using GL2 instead of SL2, but I know basically nothing there.