One question I have for 1, and I think most other community members will as well, is how it can translate into useful computations outside its own domain. In most of the polys, we have complicated algorithms, but they translate into real things, like simplifying expressions, or solving systems of polynomials. While I do think that being able to compute things about rings and modules is useful in its own right, how can you sell this to people who don't really care about this?
Yes, I see this as a problem as well. I think it boils down to what counts as "real things". (Co)homology of groups is probably a real thing to many mathematicians (e.g. hochschild cohomology computations for certain explicit, finite-dimensional graded algebras underlie Seidel's proof of homological mirror symmetry for quartic surfaces).
I think on might try to cook up "real-world" applications of group cohomology, but it seems to me that these examples would likely be contrived.
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