> Still, I don't think this works for other elementary particles, as black holes can't have color charge or weak hypercharge as far as I know (so they can't behave like quarks, gluons, W or Z bosons etc.)
I think it is expected they can. The simple reason there are no explicit BH solutions with color charge is that, in contrast to electrodynamics, there's no classic field theory for the strong interaction that we could put into our Einstein-Hilbert action.
> I think this is not even a problem in matching GR and QM, it is a problem in GR itself.
Yes and no.
All kinds of theories have singularities and infinities. Classic electrodynamics is full of them and quantum field theory is, too. Nevertheless we still say the theories are fine and treat the singularities as pretty much nonphysical. ("Point particles don't really exist / a better theory will get rid of them", "We don't see the bare particles anyway, so let's remove the infinities using renormalization", et cetera.) Yes, spacetime singularities seem somewhat more severe, but I think we have good reasons to believe (e.g. the uncertainty relations) that a theory of quantum gravity would solve this conundrum. I mean, every single singularity we worry about in GR comes with infinite curvature and/or infinite energy densities, hence necessarily requires quantum mechanics to study.
On an unrelated note: Why is no one complaining that quantum field theory, from a mathematical point of view, is completely ill-defined? It surprises me time and again that people ascribe severe issues to GR ("It has singularities", "It's not quantum") and yet completely forget that the issues in quantum mechanics (both philophical and mathematical) are much more severe. GR, at the very least, is a mathematically absolutely rigorous theory, with well-defined objects and axioms and such. QFT, in turn, to this day is a toolbox of weird "shut-up-and-calculate" heuristics.
> We can of course easily invent infinitely many solutions to this problem, but there is no way to choose between them on an empirical basis, even in principle (since we can't ever experiment with the inside of a black hole).
There is one way: Come up with candidate theories of quantum gravity and with experiments to test quantum-gravitational effects outside a black hole (there are a few ideas) and select the right theory based on the experimental results and then have the theory predict what happens inside a black hole. Boom. If you say this approach is not valid as it'll remain a theoretical prediction and we still won't be able to peek inside a black hole, you're somewhat right. But right now we're having a discussion about spacetime singularities, which are a purely theoretical problem, too. No one has ever seen them.
> GR is nonlinear, while QM is linear (if we ignore the Born rule) - so they can't describe the same system.
We already know they are incompatible but linearity has nothing to do with it. The equations of motion of interacting quantum fields are non-linear, too. In fact, electrodynamics is, too, in some sense (backreaction & self-force), and we still managed to quantize it.
> Relatedly, if applying GR to a system described by a wave function, we are not able to compute how space time will curve given that a single particle(with its mass) is usually present at many points in space-time.
I wouldn't say this is just a related problem. This is the problem of quantum gravity.
> It is hoped that solving the second problem will also solve the first, but I'm not sure this is guaranteed.
Again, I think the reason people are hopeful are the uncertainty relations. A theory of quantum gravity necessarily has to incorporate them somehow.