>Thank you for your response. What's the source of incompatibility between standard model and general relativity?
Gravity is about acceleration and flow of time. What's the problem with plugging general relativity in place of classical dynamics whenever you care about flow of time and acceleration in quantum physics? As if those gravity and other forces and particle had nothing to do with each other.

The difficulty is that the standard model uses "quantum field theory" (QFT) as the physical model in the background to describe it. However QFT is actually quite hacky in a lot of ways, the main one being that actually it predicts that every point of space should have an infinite free energy (see here for example https://en.wikipedia.org/wiki/Vacuum_energy). The way we get around this is by subtracting off this infinite energy, which is one of many similar techniques called "Renormalization" (https://en.wikipedia.org/wiki/Renormalization).

Well it turns out that if you produce a quantum field theory for gravity, it is not actually possible (or at least in any way easy) to "renormalize" it (i.e. it's really hard to subtract off these infinities)

The issue is more that QFT is really weird, rather than that general relativity is

>Why does general relativity is more incompatible with quantum mechanics than with fluid dynamics for example?

Well general relativity (GR) and QFT both describe fundamental interactions, whereas fluid dynamics describes an emergent property of electromagnetic (EM) interactions between many particles. Gravity is also usually very irrelevant in fluid dynamics unless we are looking at very large objects like the flow of matter in stars for example, so in most cases it is just EM interaction which is fully described by QFT. In other cases we have relatively weak gravity + QFT, for example in those stars, in which cases we can actually already describe the dynamics fully using approximate versions of gravity and not have any incompatibility. In cases where gravity is very strong though we do have these issues as we just don't know what happens due to lacking a proven theory of quantum gravity. One example is how the matter behaves in the super-dense core of a neutron star, known as the neutron star "equation of state", which is currently unknown, as in these cases both quantum and gravitational effects are very strong.

>My understanding is that general relativity says that gravity is not even a real force. It's just a function of acceleration caused by misshapen space-time (deformed by mass/energy).

Yes that is completely correct to our understanding, but we could maybe model it as a "real force" by writing a quantum field theory for it

>Some other things I don't understand... How gravity is considered to be a fundamental force, while Pauli exclusion principle is not, despite the fact that it can keep neutron stars from collapsing. Is it because there's no hope for finding mediating particle in the math that describes the principle because the math is too simple?

Honestly the fact that the Pauli exclusion principle isn't a force has always been confusing to me too as it behaves like one in so many situations, but it is similar to gravity really which as you said isn't really a force, but more like an influence. In fact GR says that we follow the geodesic (roughly translated to the "shortest path") along the curved space-time, i.e. it is by minimising the so-called "action". This isn't really a force, more like a fundamental law of motion. I think the Pauli exclusion principle is similar in this way, but I don't think it is a satisfactory explanation