What are the current most important theoretical problems on quantum entanglement? What is that we don't yet understand about how it works? (Not considering interpretation etc problems)
I see two current main axis of research around entanglement. I have not really followed the last years of research, so I don't know exactly which questions are open and which have been solved.
I sadly have no time to elaborate this week, but since its a community wiki, others can complete my answer.
The question is on how to characterize entanglement as a resource and how to interconvert various entanglement form into others. While the problem has essentially been solved for two-party-entanglement, many open question subsists in n-party-entanglement.
Finding quantum physics from information theoretic consideration
It is well known that quantum entanglement does not allow any faster than light communication. This is called the "no-signalling" principle. However, the no-signalling principle does not forbids theories with stronger correlations than quantum mechanical entanglement (see here.) It has been shown that if one could build such a "super-entangled" machine, one could perform various unintuitive communication-complexity related tasks (see here and here for technical papers about this, and here and here for blog posts about the second paper). The interesting open question is the following: can we find all the quantum-entanglement correlations (and the quantum mechanics formalism) from reasonable information-theoretic constraints similar to the no-signaling principle.
quantum entanglement is a basic property of multi-body quantum mechanics - or any quantum mechanics with at least two degrees of freedom - that has been understood from the mid 1920s. So for 85 years, there have been no open theoretical problems related to quantum entanglement and it has been understood that it works as well as how any doable experiment involving entanglement works even though the modern terminology related to quantum information has only been used for a few decades.