> Why would the measurement of Up/Down spin destroy information about Left-Right spin in the classical case?
It wouldn't, if you like, which would make it another thing that would be different in classical mechanics, though it's a little strained because it's assuming a lot about quantum states being like classical states.
In quantum mechanics the different spin operators do not commute, which means if you measure it in the vertical direction (and get e.g. up) then in the left-right direction, you get each of left or right with 50% probability. But then if you measure it in the up/down direction you don't get up again, you get up/down each with 50% probability, the original state is irrelevant. This is obviously different to your marbles, which just have letters that don't change.
> Can a spin be not 1 or -1? Could it be in between?
No. When you measure the quantum state you get an eigenfunction, though the state before collapse could be a superposition. Indeed, this is the case for different directions - spin up is a superposition of equal parts spin left and spin right in the horizontal spin basis, which is why you get each direction equally if you measure it in that direction.
> Doesn't that capture everything?
You're still trying to encode everything in a hidden variables theory, where the quantum states secretly know everything beforehand about what it will do, but the observer can't access the information without performing specific measurements. This immediately fails as above, because the quantum states inherently don't have fixed up/down and left/right components and each measurement in a different basis gives a random result.
You can try to fix things by adding more complex rules about the letters on the marbles changing when you read them, but it turns out there are fundamental limitations on what physical results any such theory can predict. This is famously addressed by Bell's theorem (as in, seriously famously, this is really important), which demonstrates specific limitations on what physical results are allowed by such models, and experiments have shown that quanum mechanics does breach these limitations. I'm not sure how to explain more simply what's going on though.
(Strictly, we can allow hidden variables (though more complex than a predetermined left/right and up/down as above) by breaking the speed of light to let the separated states communicate instantly, but this brings up its own problems.)
Sheesh, these are awfully handwavy questions for someone who was in a math ph D program. I should really sit sown and learn quantum mechanics for a while with the math. How long do you think it would take to reason intelligently about it? Going by what Feynman said, maybe I can't.
Bell's theorem remains true, but pilot wave theory achieves the quantum results by sacrificing locality instead of hidden variables.
> how does the description of entanglement as "spooky action at a distance" not violate those?
It's just a name, and an old one at that. I suppose if you ascribe it to non-locality then it really does involve something FTL on some level, but even with a standard locality-preserving, no hidden variables theory, no information travels faster than light so you don't actually hit a FTL problem. The strange thing is that the quantum state appears to behave consistently regardless of its spatial extent, which is weird.
> How long do you think it would take to reason intelligently about it?
I would have thought that someone in a PhD mathematics program would be well equipped to understand the mathematics - maybe the hard part is finding a good resource that takes things in a good order. I'm afraid I can't really recommend anything, though Feynman's stuff is probably good.
Suppose Alice and Bob had unlimited pairs of marbles labeled "RU" "RD", "LU" and "LD" paired up with their opposites respectively, e.g. RU with LD. Alice would randomly select four marbles, one from each pair, and Bob would select the corresponding opposites. Now they separate and go to their homes, not knowing what the marbles are.
Alice checks the first letter on the marbles only, and sees that she has R* and R*. She concludes that her spin is "LEFT" and that bob has the "RIGHT" marbles. Now, she decides to check all the letters on her marbles and realizes she has, say, two RT marbles. She knows that Bob has two LD marbles.
Can a spin be not 1 or -1? Could it be in between? That would be the case here if Alice had RU and RD, and once she checks both letters, she knows what Bob has as a result.
But in the classical case discovering first letter doesn't give you any information about the other. Doesn't that capture everything?