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It is still true that there is a surjective algebra map $U(\mathfrak{sl}(V))\to D(\mathbb{P}(V),\mathcal{O}(n))$ for essentially the same reason as for the full flag variety and the case $n=0$. However I don't know if there is a good description of generators of the kernel.
Do you mean that $R$ is a skew-polynomial ring in the sense of en.wikipedia.org/wiki/…, so that typical elements are of the form $\sum_{i=0}^n \lambda_i F^i$ with $\lambda_i\in \Lambda$? I think you must but wanted to check.
It isn't a purely algebraic statement since you can't define the complex numbers in a purely algebraic fashion. The complex numbers are much bigger than an algebraic closure of the rationals.
Are you assuming that $G$ is finite or something? Otherwise $G=\mathbb{Z}$ and $M=k$ with the trivial action seems like an easy counterexample to Question 1.
I was a bit sloppy in my last comment. As noted by Qiaochu Yuan below the outer Hom should be of right R-modules with the left action of R coming from the left R-module structure on R and the right module structure coming from the left R-module structure on the inner Hom(B,R).
