What goes wrong when there's torsion?

In my novice understanding: I think the usefulness of studying cohomology of coherent sheaves on a scheme is primarily for analogy (although there are bound to be some direct applications of the theory). Coherent sheaves are the class of sheaves where cohomology works 'nicely' in the setting of sheaves of modules on a (separated and Noetherian, say) scheme. There are analogous classes of sheaves ($\ell$-adic constructible sheaves, etc.) for the étale cohomology which are the 'nice' ones in that setting. Many similar formal properties hold, and the derived functor formalism unites the two.

good point! If you'd like to put that up as an answer, I'd be happy to accept.

Thanks for the update! This seems to be getting to the heart of the matter. One more question - it seems that the definition of names, being internal to $M$, should work because $M$ "thinks" it's well-founded, but in defining the values, we need to either work with "real" set-membership (which is well-founded) and end up with something not having anything to do with the inductive construction in $M$, or attempt to induct on $M$-membership, but this time in the "real" world, where this isn't well-founded anymore. Is this correct?

This looks great! So that there is an explanation on this site, would you be able to explain a main idea or two of this paper here?

Thank you for the answer! I suppose I wasn't very clear in my question, but I was really looking for a more narrow answer - why don't the arguments for forcing over a countable transitive model (so no need for syntactic or Boolean-valued notions) hold when the model is just countable but not necessarily transitive?

Even though the map isn't a fibration, the sheaf cohomology is fairly straightforward in this case. Since all the fibers are circles, the $i^{th}$ derived pushforward sheafs vanish for $i$ not $0,1$ (so the spectral sequence degenerates and gives an exact sequence looking a lot like the Gysin sequence for an $S^1$-bundle). The $R^1$ sheaf is the subsheaf of the constant sheaf $\mathbb{Z}$ whose stalks at a point $x$ are $m\mathbb{Z}$ with $m$ the multiplicity of the fiber above $x$.