Resolving questions aside, can someone kindly explain me why this question deserved so many downvotes? Is it "too dumb" (not at the level of professional mathematicians) to be asked here, and preferably asked at Math SE instead?

The chief example to deal with would be $M=\mathbb C^n$ though, and thus I need noncompact $M$ too.

I see, thank you.That's one less weight off my heart (I always thought that I don't understand sheaf cohomology very well because I don't understand how to think of the canonical resolution using discontinuous sections).

This somewhat gives me a peace of mind. Looks like I can consider higher sheaf cohomology in at least two ways: 1. Literal invariants (for Dolbeault cohomology, for exmaple) 2. extra baggage that tags along for Euler characteristic (for Hirzebruch-Riemann-Roch and coherent cohomology) quite much like how physicists mostly care about velocity and acceleration.

I suppose you are talking about the "canonical resolution" where the sheaf of discontinuous sections are taken interatively? I hope I don't sound like too much of a complainer but at second or third iteration already I found the resulting sheaf (and the cohomology) quite unintuitive... even though choices of the acyclic sheaves were canonical here.

Also, I'm a little ashamed to ask this, but I never really understood the point of considering conjugate-differential $d\bar z$ (I know that $\bar\partial f=0\iff $f is holomorphic, but still don't really see what's the point of $d\bar z$). Could you tell me more about what makes their integration meaningful?

Indeed I was interested in sheaves like this (holomorphic forms, etc.) but I'd say that generally I am not interested in structure for the sake of structure. I mean, self-intertwining outgrowth of abstraction is always an exciting side of mathematics, but I'd still say "that doesn't have much of a point" if they didn't point at something outside the formalism. I found Will Sawin's answer very helpful because it was pointed out that manifold classification problem gets a huge help from cohomology, which is a problem that can be posed independently of any cohomological concerns.