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.