Edited, I don’t believe $P$ is necessarily graded, since it can have saturated chains of different lengths. However if you know this setup in the graded case, I would be interested.

Sorry, I'll edit the question, I used covering forgetting the technical meaning, I just mean a nonconstant map of curves, possibly with ramification. I'll think about your comment further, though I don't see how the noncyclic case works, but that's probably because I'm not comfortable with $Pic^0$.

Yep, this is all taking place in the universal enveloping algebra, noting that the canonical map from $L$ to $U(L)$ is injective, so we can verify stuff there.

I don't have a reference unfortunately, only remember reading it on here.