I think that by using hypercoverings instead you should get all local schemes in the limit. I don't see that using Nisnevich coverings instead would change anything, the tensor product of two Henselian local rings will still not be local I believe.

You can not in any obvious way compute even the cohomology of the tensor power. The Künneth theorem is about the action of $G^n$ on $V^{\otimes n}$ not about the action of $G$.

You have to pass to $\mathbb Q$ to get the projector.

I think that in our enlightened age we should be allowed to disregard equestrian prejudice against the horseless so I stand by my comment.

I know you are not the one making up this particular piece of terminology but to me it seems a pedestrian should have ample time to look around and make everything precise whereas a driver swishes through at high speed and should not...

Now that I have looked at it Gille-Szamuely is indeed a proper reference. I kind of expected there to be a reference but just enjoyed myself with figuring things out and felt that I might as well share...

Add to this that this loop is homotopically non-trivial even when passing from $SL_2$ to $SL$ which seems to be the OP's question.

If $G$ is compact so that $H$ splits up as a topological sum of isotypical components, then each isotypical component has to have infinite multiplicity. Under that assumption it seems to me that one can apply Kuiper's result to each of the components.

My recollection is that it wasn't. In any case I have done exactly this and it worked (though you may be right that it is not formally guaranteed to).

You have a window of opportunity of a few hours when you can still edit your submission before it is put into arXiv. During it you will know the arXiv id of tha article. Hence you can submit both of them,get their ID's and then insert them.

Just take the square root of an element of $K$ with valuation $1$ at the prime.

@David: The $\mathbb F_{p^n}$'s are not linearly ordered, containment is given by divisibility of the exponent $n$.