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Have you seen Quantum geometry from phase space reduction, by Conrady and Freidel, at arxiv.org/pdf/0902.0351.pdf? They discuss the space of tetrahedra with given face-areas, and pages 7-9 have some formulae that look related, at least if I squint hard enough.
No need to assume unitarity. Smooth reps of $SL_2(Z_p)$ are finite-dimensional and factor through $SL_2(Z / p^k Z)$ for some $k$. Branching from $G$ to $K$, these $k$'s are unbounded. For Shalika's thesis... I'd consult Nevins' papers for the reference.
Yes. Your two definitions of $C^\infty(G({\mathbb A}))$ are equivalent, and with a compact support condition, equivalent to the third. The exercise is to prove that, given a closed embedding of $k$-groups from $G$ to $GL_n$, the subspace topology on $G({\mathbb A})$ from $GL_n({\mathbb A})$ coincides with the restricted direct product topology, with respect to the open compact subgroups $G(K_v) \cap GL_n(O_v)$.
It looks like you've got everything right, and the answers to your questions are uniformly YES. A standard reference is Borel and Jacquet's article in Corvallis (Proc. of Symposia in Pure Mathematics, vol 33 (1979) part 1, pp.189--202), and they give references to earlier work of Harish-Chandra and others along the way. See, for example, section 1.8 (working with real groups) and 4.4-4.8 for the adelic formulation.
