In Bell's inequalities, what is the meaning of assuming that the physical properties $P_Q,P_R,P_S,P_T$ have definite values?

This, in a nutshell, is the whole issue of quantum. You say "of course an electron will have either spin up or spin down". It might seem like it should, but this is founded purely on your intuition about what the world around you is like. You have no a priori law of the universe that rules out other options. And hence, when an experiment shows you that assumption 1 does not hold in certain scenarios, you have to rip up your intuition and start from the beginning. You can no longer say that an electron is definitely either spin up or spin down.

Actually, an electron will only have spin up or spin down when we decide to look at it. Generally, quantum mechanics postulates that the state of the spin exists in superposition between up and down, only collapsing to a definite value--and becoming an element of reality--when we perform a measurement. In the first assumption you cite, when the physical properties are assumed to have definite values, we mean that they really, actually, objectively have those values in nature. When we measure, we just obtain those values, which would have existed regardless of our decision to measure.

The weirdness of quantum mechanics lies in the fact that our measurements do not simply obtain this information, but could actually affect reality itself. Mathematically, we say that when we measure, the system will collapse to an eigenstate of the operator that we are measuring. When two operators commute, they share an eigenbasis--unfortunately, position and momentum do not commute. If we measure the position of a particle and obtain some value, $x_0$, we can be certain that right after the measurement, the particle has been localized to $x_0$. The momentum $p$, however, is now indeterminate, and we can only describe it with a probability distribution (i.e. the norm-square of the wavefunction in the momentum basis) over the values it could take if we were to measure the momentum. Needing to talk about a probability distribution, as opposed to a definite value, means that momentum cannot be an element of our reality anymore. And of course, if we did measure it, the position of the particle would become indeterminate instead.

Clearly, whether we choose to measure $x$ or $p$ affects the reality we observe, which is completely opposed to assumption (1) and the intuition of the authors of the EPR paradox, who took this indeterminacy as a sign that quantum mechanics was incomplete, that the wavefunction could not accurately predict all elements of reality simultaneously. With experimental verifications of Bell's theorem, it is apparently the case that not all such physical properties ($Q,R,S,T,dots$) can be elements of reality at the same time, so it's no wonder that quantum mechanics cannot definitely predict all their values. You could instead reject assumption (2), but then you are assuming that information can travel faster than the speed of light.

The "realism" assumption basically amounts to saying "the system's state can be described by a probability distribution, operations on the system can be represented by probabilistic transitions, and here are some specific distinguishable states that I know the system can be in".

An example of a system that doesn't have this property is one where an operation like "transition to the state that currently has the highest probability assigned to it" is available. There is no stochastic matrix that corresponds to that operation, so it is not a probabilistic transition. Alternatively, the initial state could have an invalid combination of probabilities like "110% Off and 20% On". Really the space of "things that don't behave like probabilities" is quite large.

One way to think about this is to realise that the "physical properties" in QM (but also more generally in physics) are not just a function of the system under examination, but also of the measurement apparatus that you are using to probe said system.

In the case of QM, you cannot observe the system without significantly perturbing it, as you do in classical physics. It therefore becomes tricky to even define what "the state itself" even means, as all we can do is characterise how it responds to the possible measurements we perform on it.

Consequently, there might be different ways of measuring a system which are incompatible with one another, meaning that the act itself of performing measurement $A$ will affect the results of a future measurement $B$. In such a case, what would it even mean to say that "$A$ and $B$ have independent values independent of observation"? If measuring $A$ changes $B$, clearly they cannot have simultaneous definite values.