You might also be interested in projecteuclid.org/euclid.dmj/1092749255 which gives a derived version of the same theorem and is in English.

where continuity means as in your edit.

The paper I mention below shows that there is no counter-example in the continuous case. I don't know what happens without continuity.

If you can see MathSciNet mathscinet.ams.org/mathscinet/search/….

Do you have any continuity constraint on sheaf endomorphisms?

Surely no because $R$ may be Noetherian and $I$ may be infinite whence the injective envelope of the direct sum of the $Q_i$ is the direct sum of the $Q_i$ which is not the product of the $Q_i$. Or did you mean to insist that $R$ be non-Noetherian.

You may also find books.google.co.uk/… to be of some interest re your final question.

I think that consideration of finite groups (which can be viewed as algebraic groups of course) might help you assess whether anything like this is at all plausible.