swansont

They didn't cause the particles to spin - they already had spin. What they did was measure one particle, and instantly knew, because the particles were entangled, what the spin orientation of the other particle had to be. This is because the spins had to add up to some value.

Your plasma ball is giving off X-rays. Very "soft" ones (i.e. relatively low energy), to be sure, so I'm a little surprised that it sets off the Geiger counter, but I suppose it's possible.

Not quite. C-14 decays to N-14. It's a beta decay, so a neutron changes into a proton, and emits an electron and antineutrino. For C-14 to decay to C-12, it would need to emit two neutrons. Neutron emission can happen for some nuclei, but that requires an large excess of neutrons, and doesn't happen in this case.

In a medium (n>1) the wavelength becomes lambda/n. The frequency stays the same. Light travels at c/n through the medium.