I will attempt to answer questions 1-3 as best I can. The others, I defer to the other excellent answers here provided.
Before I start: "The inequalities between the two hypotheses rise when the particles are measured on axes which are between 0-90 degrees off axis from each other, correct?" -- correct.
2. What are the predictions made by local realism?
I think that this is really the crux of the problem, regardless of what the predictions made by quantum mechanics are. This is because Bell's Inequality does not state a prediction of QM -- it states a prediction of local realism (or of other sets of closely related philosophies, such as locality + counterfactual definiteness) -- and there is lots of evidence that the prediction of local realism made by Bells' Inequality does not hold. Thus, what QM predicts is only relevant if you are interested in one of its many interpretations to replace local realism. Of course, via these same experiments, the results tend to match the predictions of QM, so they also provide evidence for the QM equations, but I think that this not the main purpose of Bell's Inequality.
Bell's Inequality is a very abstract statement designed to cover any local realism theory. So, since intuition is what we are after, allow me to propose a particular local realism theory, which the experiments for Bell's Inequality will therefore provide equally good evidence against:
Hypothesis 1: The spin of a particle is governed by a hidden variable $\theta \in [-\pi,\pi)$. Denote $\theta_\phi$ to be the outcome of measuring a particle's spin when our measuring equipment is calibrated at an angle $\phi$ (so for all $\phi$, $\theta_\phi = 1$ or $\theta_\phi = -1$). In other words, even though we always measure spin to be up or down, there is some "hidden variable", $\theta$, that is a continuous-valued spin that is the "real variable" which is the "true" spin, we having only a poor window with which to view it, namely, $\theta_\phi$. For the sake of concreteness, we hypothesize the following behind-the-scenes mechanism for our measuring equipment:
$$\theta_\phi = \text{sgn}(\sin(\theta-\phi))$$
(this is a square-wave by the way... in a sense, you just round the angle you are measuring relative to the angle of your equipment. e.g. if $\phi=0$ and $\theta$ is negative, you get "down" and if the $\theta$ is positive you get "up")
Note that Hypothesis 1 gives us a mechanism for local realism since a particle has a definite spin (realism) given by $\theta$. Also, an explanation for correlations between measurements when we study pairs of particles at a particular "angle" can now be explained by a local property, $\theta$.
The next few paragraphs operate under the assumption of Hypothesis 1.
Now, per the usual example, let's generate sets of pairs of particles with opposite orientations. Let's just focus on a handful of pairs that we managed to generate, and let's pretend that we can peek under-the-covers to see: $\{(\theta_1, \theta_2)\} = \{(\pi/4,5\pi/4), (4\pi/3,\pi/3), (0.001, 0.001+\pi)\}$. We want to think about what happens when we measure these particles. Calibrate detector A at $\phi=0$ and detector B at $\phi=\pi$. If we split up the pairs, and send $[\pi/4,4\pi/3,0.001]$ to detector A, and $[5\pi/4,\pi/3,0.001 + \pi]$ to detector B, what do we expect to get? Plugging stuff into the above formula, we expect to get $[1, -1, 1]$ at detector A and $[1, -1, 1]$ at detector B. Play with the calibrations at detector A and B and re-plug stuff into the above equation. Note that no matter what you set them at, as long as they are opposite ($\phi_\text{A} - \phi_\text{B} = \pi$), then we get identical results at detectors A and B (although perhaps not the exact sequence $[1, -1, 1]$, depending on the calibration).
Now, notice that if we change the calibration of only A by a very small angle $\phi_\text{A} = 0.002$, then the value of the third particle in our list $a_3$ at detector A will flip. The corresponding measurement at B should not, because we have not altered its calibration, and $\theta$ for $b_3$ remains the same. In other words, if we don't change the calibration of B $\phi_B$, and we don't change any of the $\theta$ of our particles, then it is of no consequence what is happening over at A, whether researchers are measuring particles, or whether they have all gone to have a beer, what we measure at B is totally unaffected and has only to do with the calibration and particles at B. This statement is a necessary condition for local realism to hold. If, somehow, the corresponding measurement at B changes depending on whether an observation occurred at A, then either it communicated with its pair particle over at A (to change its $\theta$), or some other implicit assumption of Hypothesis 1 has fallen through. So, one prediction of our Hypothesis 1 is that the measurement for $b_3$ remains the same whether or not we make a measurement at $a_3$. If we can show that this does not hold, then Hypothesis 1 does not hold.
The exact situation when we expect Hypothesis 1 to fail, due to the predictions of QM, are kind of strange. If we measure particle 1 at B, then its partner particle 2 at A, and the re-measure particle 1 at B, QM does not expect the measurement at B to change. We only expect the measurement at B to be "altered" if we look at A first. This makes observing the supposed "alteration" difficult!
However, Bell proposed the following experiment by which we can test Hypothesis 1 (and a whole class of related hypotheses). If we generate a boatload of particle pairs according to a common general scheme, and then re-calibrate A and B to various convenient values, we can predict the probability of various observations at B both with and without having "looked" at the particles at A.
Here's the setup: Generate a very large quantity of particle pairs with the first particle having uniformly-distributed $\theta_1$, and the second having an opposite orientation $\theta_2 = \theta_1 + \pi$. We can test uniformity by simply calibrating our measuring apparatus at random locations and making sure we get an approximately equal number of "ups" and "downs". The only way for this to happen is if the $\theta_1$ are uniform. We can test that the two particles are always opposite by checking that, when A and B are calibrated at $\phi_\text{A} - \phi_\text{B} = \pi$ apart from one another, we always measure identical readings for each particle in a pair. Set $\phi_\text{A} = 0, \phi_\text{B} = \pi$. Alter $\phi_\text{A}$ (and only $\phi_\text{A}$) by a little bit. Generate another bunch of particle pairs. Now, some quantity of the particle pairs will not produce identical measurements (like our $a_3, b_3$ above). Write down this quantity $x$. Just to double-check, reset $\phi_A$, and alter $\phi_B$ by that same angle. Generate a bunch more particle pairs using the same mechanism. You should see that the number of unequal measurements is approximately $x$, because the situations are symmetric (but not exactly equal, because our $\theta$ are random). Just to quadruple-check, do this a whole bunch of times to convince yourself that the number of unequal measurements is pretty much always around $x$.
Here's the expectation: Now, change $\phi_\text{A}$ and $\phi_\text{B}$ by that small angle. In order for the problem to appear, we need to consider what might have been had we not altered $\phi_\text{A}$ or $\phi_\text{B}$ or both. If we hadn't altered either, because the measurements are all governed, under-the-covers, by $\theta$, we would have measured identical values for all pairs. If we had only altered one or the other, we would have measured different values for $x$ pairs. If we alter both, even if none of the pairs that "change" overlap, we measure different values on $2x$ pairs. Namely, all of the pairs whose measurements "changed" at B plus all of the pairs whose measurements "changed" at A. For the remaining pairs, since their measurement didn't change at A and didn't change at B, they still give identical measurements. If there is any overlap in which pairs flipped measurements at A and B, then the number of pairs giving different measurements will be strictly less than $2x$. To reiterate, this expectation only holds if the measurements at A and B do not effect each other. It also only holds if it is meaningful to speak of "what might have been". If the simple act of observing the spin of particle 1 at A changes the value of $\theta_2$ of its partner at B, then the situation where we make measurements at A and B need not have this particular relationship to the situation where we make only a measurement at A. For example, the act of measurement at A could change all of the $\theta$s at B to be totally random. Or it could change the $\theta$s at B to be the number predicted by QM. The only important thing here is that if A and B "talk", then the number of "different" measurements might be $>2x$.
At this point, it is worth noting that the exact mechanism we proposed above is irrelevant to the argument as a whole. You can replace all of the talk of "$\theta$" and the mechanism we proposed by which it is measured by talk of some "arbitrary locally-real variable encoding the spin info" and the inequality still holds.
1. What are the predictions made by quantum mechanics? & 3. How do they differ?
Basically, QM predicts that for certain calibrations of the equipment at A and B, we will reliably observe $>2x$ pairs that now give different measurements when we alter both $\phi_\text{A}$ and $\phi_\text{B}$. How much different depends on complex maths that are above my pay grade. If anyone in the community has a link to a location with an explanation of this math, please comment and I will edit it in.
However, as I said above, it's largely irrelevant to Bell's result what those predictions are. Simply performing the experiment and noting that the number of pairs with different measurements is $>2x$ is enough to reject local realism, even without anything to replace it with.
Somewhat orthogonal to the predictions made by QM are the available interpretations of this result now that local realism has been tossed. This answer to a related question provides a discussion of how these interpretations tie into the results from Bell's inequality.
