I have not read it, but I think the article Heinzner-Loose, "Reduction of complex Hamiltonian G-spaces" (1994, link) may be the original reference for your first question.

In the compact (but not necessarily projective) case, both questions are treated in Fujiki, "Kähler quotient and equivariant cohomology" (1994, link).

I would also be interested in seeing a reference in the non-compact case -- let me know if you find one!

I have not read it, but I think the article Heinzner-Loose, "Reduction of complex Hamiltonian G-spaces" (1994, link) may be the original reference for your first question.

In the compact (but not necessarily projective) case, both questions are treated in Fujiki, "Kähler quotient and equivariant cohomology" (1994, link).

I would also be interested in seeing a reference for the second fact in the non-compact case -- let me know if you find one!