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Quantum entanglement is the mechanism by which quantum correlations between two sub-systems survive even after being physically separated from an interaction region. The correlations could in principle survive without neither time nor space constraint.

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Quantum entanglement, how do we know there was no spin?

The answer is Bell inequalities. https://en.wikipedia.org/wiki/Bell%27s_theorem Experimentally either Bell's theorem is true or there are non local (read: faster than light) communication of the "tru …
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How does a unitary operator acts on an entangled $|\psi\rangle$?

Your unitary looks like a controlled-X from the second to the first qubit. It is fairly easy to work through what it does by just plugging in definitions. $U = |00\rangle\langle 00| + |11\rangle\la …
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