How do we know for sure that the photon's orientation isn't determined until we we measure it ?
That's what Bell's inequality proves. The assumption that the photons have definite polarizations (we just don't know what they are) is a local hidden-variable theory, which Bell proved cannot reproduce the predictions of quantum mechanics.
If it's predetermined then no need for spooky action at a distance. So why the spooky action at a distance issue in Physics ?
Actually we don't know it. We just know that photon's polarization (if it has such a property) can't be the only thing that determines measurement results at different angles.
Bell proved that the results are NOT predetermined (under certain assumptions).
Look at this post: https://www.physicsforums.com/threads/a-simple-proof-of-bells-theorem.417173/#post-2817138
There isn't really a spooky issue in physics. It's just that you can interpret the Bell violations in various ways by rejecting different assumptions. Most people reject the non-contextuality assumption, since we know that it must be rejected anyway (for different reasons), but you can also reject the locality assumption (together with the non-contextuality assumption). That's what the hidden variables advocates do.
Yes. We have tried to explain this a number of times already in this thread. The reduced density matrix of the EPRB state is the (normalized) identity matrix. All the difficulties you are having in this thread (like partial traces and matrix multiplication) are really linear algebra difficulties and not quantum mechanics difficulties. Teaching these basics through an online forum is quite cumbersome, so I suggest you pick up a some introductory linear algebra textbook (for example Halmos FDVS) and then come back with specific questions.
Actually we don't know the value of any observable until we measure it.
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