If you cannot read the paper but another paper explains what is says can you not adopt the phrasing 'x records in [1] that y proved in [2] that...' or something similar.

I think it would help if (a) you were more explicit about how you define $F$ (b) you explained what you mean by reconstructing the algebra from the functor. I think I can guess the answer to both questions but it is hard to be sure I'm right.

Perhaps best of all would be to show inductively that the induction hypothesis is true for $k$ at most the minimum of $n$ and $m$ and then observe if the result is true for $k=n<m$ then we have a contradiction. I feel the idea is getting lost in the details now. But I won't change it.

Also even if it should be twice as long as it is now it still need not be described as very long.

Fair enough. I really just edited it to make it better than it was before. I think it is now easier to improve upon without completely rewriting. The proof shows that if we get to $k=n<m$ then the $v_i$ are not LI. I agree that could be made more explicit. To be honest I was seeing the conclusion that $m\leq n$ as a corollary of the other part.

Does the proof really have to be so long? See the Wiki entry now. It could still be slightly lengthened for maximal clarity but I don't think it is so hard to write out a shortish intelligible proof.

The answer is that it is false. For example if $G$ is the trivial group then the projective dimension of $\mathbb{Z}$ is zero but the injective dimension is one.

I think that all these things will be true when $G^p$ is profinite (in fact necessarily pro-$p$ in that case). Moreover $G$ finitely generated will suffice for this. I suggest consulting a book on profinite or pro-$p$ books such as those by J.S. Wilson or Dixon, Du Sautoy, Mann and Segal.

I think Torsten has now basically answered that question. Probably never yes for infinite groups but you can say something.

I should perhaps add that these $I$-adically completed things are studied mostly under the name of Iwasawa algebras. See math.uiuc.edu/documenta/vol-coates/ardakov_brown.html for a survey of algebraic results.

I think this is the same as the question 'Why is one not prime?' It cannot be written as the product of two proper factors.

The answer to your first question is yes. If you differentiate the action of the group $PSL_2$ on the Riemann sphere at the identity to give a Lie homomorphism from the Lie algebra $\mathfrak{sl}_2$ to vector fields on the Riemann sphere. The image of this map on a coordinate chart obtained by removing a point at infinity is your Lie algebra of complex quadratic vector fields.

Oh. Rereading I see this isn't quite what you ask for. But I'll leave it anyway.