@VladimirDotsenko I thought that substituting $B^e = A$ and pulling back by $\mathrm{Der}(B)\to \mathrm{Der}(A)$ recovers enough information for my computation, but you're right. I should've taken $A^e/[A^e,A^e]$ for better formulation. Thanks.

Wow, I didn't know that. Thank you so much!

I'm sorry; I think that's not necessarily the case (at least in general). Even if $X\to V$ is an injection for a set $X$ and a vector space $V$, this may not induce an injective linear map. Are there other reason that I'm missing?

Thank y'all for the answer and a counterexample. Under the residual torsion-free-nilpotency, is it also true that the map $k[G]\to\widehat{k[G]}$ is also injective?

Sorry, but it's not clear to me. If $f:M\to N$ is an epimorphism of graded f.g. modules and an iso modulo the maximal ideal, $\mathrm{Ker}f + H = M$ implies $H=M$ for a graded submodule $H$ by graded Nakayama's Lemma, but I'm not sure for a general submodule $H$. Am I missing something?