"Anyone who is not shocked by quantum theory has not understood it." - Niels Bohr
“I think I can safely say that nobody understands quantum mechanics.” - Richard Feynman
Feynmann disagreed with Bohr as to whether Bohr understood quantum mechanics*. But don't imagine for a second that this means Feynmann wasn't shocked by quantum theory too. Feynmann was very much in the "This problem is too hard for current science." camp.
(* At one point Bohr specifically asked for Feynmann to be his assistant because Feynmann was the only student willing to tell the great man he was wrong.)
And I say no, the experimenter does not know what is in the other box.
The box and what was in it may be nothing but ash light years away.
But "light years away" is a theoretical extrapolation. In the actual experiments the boxes are a few meters to a few kilometers apart, and the experimenter does know what's in the other box, after the fact when he compares the outcomes.
There are a couple things about the glove analogy that are seriously misleading. In the first place, there's only one possible outcome for a given glove. If it's a left glove it's a left glove no matter how you look at it. That's not how it works in an entanglement experiment. If you shoot a laser through a barium borate crystal, sometimes a photon in the laser beam will "parametrically down convert", which means it breaks up into a matched pair of photons that always have opposite polarizations; these are what the gloves are supposed to be analogous to. But the polarization of a photon isn't like the handedness of a glove. Whether a photon is polarized one way or the other depends on which way you look at it. To make the glove analogy fit, you'd have to use a peculiar type of glove that's manufactured in the dark and that can be a left glove if it's first illuminated from the north or the south but a right glove if you first illuminate it from the east or the west. When we say the two photons always have opposite polarizations, that's only when we measure them both the same way. If we measure them the opposite way then they'll always have the same polarization -- it's as though if I open my glove box from the south and my friend across town opens hers from the east and then we compare notes, it will always turn out that we had two left gloves or two right gloves.
Now so far that's merely weird, not crazy. You can do all that with smart gloves that sense which direction the light is coming from and shape-shift in response. You manufacture one smart glove with the rule (north/south=left, east/west=right, northeast/southwest=left, northwest/southeast=right, north-by-northwest=right, south-by-southeast=left, etc.). You manufacture the other glove with exactly the opposite rule (north/south=right, east/west=left, northeast/southwest=right, northwest/southeast=left, north-by-northwest=left, south-by-southeast=right, etc.). That way whenever the two boxes are opened with the same orientation you're guaranteed to get a left glove and a right glove. Whenever the two boxes are opened ninety degrees apart you're guaranteed to get either two left gloves or two right gloves. It all works by prior arrangement, so there's no need for the two gloves to talk to each other after they're separated. This is what the "Take a pair of gloves and put one of them in one box and the other in another box without knowing which is in which box then ship one box far away" analogy is intended to be analogous to -- all that complexity about rules and shape-shifting gloves hasn't changed anything fundamental.
So here's the part where physics gets crazy, and here's the second thing about the glove analogy, the point where it gets seriously misleading. The "One box will have a right hand glove and the other a left hand glove. ... When one box is opened and observed, the experimenter immediately knows what glove is in the other box even though it may be light years away. Not at all mysterious." bit can evidently only be referring to what happens when both experimenters open their boxes from the
same direction.
But that's not what people do in an entanglement experiment! The whole bloody point of doing these experiments is to find out what happens when the boxes are opened in different directions! We already know the gloves will be opposite if we open the boxes along the same axis. We already know the gloves will be the same if we open the boxes along perpendicular axes. What we don't know, what we intend to find out, is whether the gloves will be the same if we open the two boxes in directions
45 degrees apart from each other, or 22 degrees apart, or whatever odd angle.
The crazy thing about the real world is that how often you get identical photon polarizations and how often you get opposite photon polarizations depends on the angles you measure the two photons from, and there's a formula for how often it happens at 22 degrees and at 45 degrees and so forth -- a formula predicted by quantum theory and confirmed by experiment -- and it turns out to be a formula that cannot be accounted for by prior arrangement between shape-shifting smart photons. That's why smart people who've studied quantum physics believe so many weird-ass counterintuitive proposals like consciousness affecting particles and parallel universes and whatnot. So the glove analogy amounts to
Bohr: Look at the crazy thing that happens when the measurements are 22 degrees apart! How do you account for that?
Einstein: Nothing crazy happens when they're 0 degrees apart. See how sane the universe is?
It's the physics equivalent of
Gnome: Step 1 is Steal underpants. Step 3 is Profit!
Kyle: What's step 2?
Gnome: Step 3 is Profit!