Your definition does not make much sense to me: if $I$ is a proper ideal of $A$, then $V\otimes_kI$ is non-trivial $G_A$-stable submodule of $V\otimes_kA$. The standard definition assumes that $V\otimes_kA$ is irreducible for all field extensions $A$ of $k$; but the fact that every irreducible representation is absolutely irreducible is standard, since $k$ is algebraically closed

What's an example of such an $R$?

It seems to me that $f_*\mathscr{F}$ is locally free as a $f_*\mathscr{O}_X$-module if and only if $\mathcal F$ is locally free as an $\mathcal O_X$-module.

You can't prove it, because it is not true, for example, for a non-Galois connected étale cover.

Given an element $L\in Pic(X\times Y)[2]$, the associated map $X to Pic(Y)[2]$ is constant.

The "local ring of an irreducible component" is the stalk of the structure sheaf of an integral scheme at the generic point. This is, in fact, a field.

You are trying to lift a holomorphic $\mathrm{PGL}_{k+1}$-principal bundle to a $\mathrm{GL}_{k+1}$-principal bundle. Using standard methods in bundle theory, this reduces to the statement that $\mathrm H^2(\mathbb P^n, \mathcal O^*) = 0$, where $\mathcal O^*$ is the sheaf of invertible holomorphic bundle. From the exponential sequence this reduces to the statements that $\mathrm H^2(\mathbb P^n, \mathcal O) = \mathrm H^3(\mathbb P^n, \mathbb Z) = 0$, which are standard.

What do you mean by the "coarse space"?

Yes, sure, the projection is also defined by real polynomial maps.