Is it possible that Bell inequality tests provide experimental evidence in support for the EPR claims in their 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?", where Bell tests eliminate local realism as cause for entanglement, but allow for correlations which may be non-causal? In short, is it possible that quantum entanglement is just simply non-causal and therefore will never offer an opportunity to allow communication?
Yes - these experiments have been conducted, most famously by Aspect et al., but also by others, see Wikipedia. They all observed violations of Bell's inequalities - Our world is therefore not local-realistic in the sense of Einstein.
An extension of Bell's inequalities by Legett (Legett inequalities) holds for non-local realistic theories. Their violation has been observed as well and therefore also ruled out this particular version of "realism".
Independent of that, it is impossible to send information via quantum entanglement by the No-communication theorem. In essence, the problem is that you need additional information in order to make sense of the observed states of the entangled particles, this information however can only be transmitted via classical channels.
Apart from the 'yes' as the first word in @Daniel's answer, I think the rest of his answer is correct. EPR claimed some 'spooky action at a distance', and that it proved QM was not true. Bell's theorem and inequalities, and the experiments carried out, proved that there was no such issue, that QM does not allow a local hidden variable explanation, by carrying out objective experiments in various ways equivalent to the intent of EPR. Yes or no depends on which exact words one uses. To be more specific and exact, see below.
Over the years there have been many experiments testing Bell's theorem and inequalities. They all concluded that as Bell's theorem claimed, quantum mechanics predicts different results than the assumptions of local realism would predict, and those experiments showed the quantum mechanical results. Local realism simply means two things: one is locality, or causal non-instantaneous propagation of information; the second, realism, is maybe better known by physicists as the existence of hidden variables that then determine the results of a quantum mechanical system. So those experiments showed that local realism is not a possible alternative or way to explain quantum mechanics.
This was actually the beginning of quantum information theory (and a lot of experiments and so on).
But those experiments all had some loopholes, things could get pretty tricky in a way that is best to read directly or through summaries. The Wikipedia article referenced below summarizes most of it. It also mentions and references experiments carried out more recently, as recently as December 2015 (or maybe just published then), that were loophole-free, and this claimed to resolve the issue once and for all (as suspected by most physicists, and shown, though not totally conclusively, in many experiments). The set of results referenced claimed over 5 sigma certainty in the results.
See the Wikipedia article for some of the loopholes and the reference to the loophole free experiments of 2015. https://en.m.wikipedia.org/wiki/Loopholes_in_Bell_test_experiments
By the way QFT is inherently local, and with no hidden variables (i.e., non-realist in the sense used in these papers).
Due to the work on information theory, it is now pretty well understood and accepted also how the related issue of wave function collapse takes place as due to (or is really) decoherence through interactions with the environments (whether you measure or not), and also well understood as the macroscopic approximation when many interactions with the environment occur such that a microscopic QM description gives way to the macroscopic description. In QM this is also well described and formalized using density matrices.
Before Bell's theorem and the experiments the EPR paradox bothered a few physicists, but mostly it was ignored. It is now known, and has a firm footing, that yes there is entaglement, but it does not really mean any spooky action at a distance (in the sense of Einstein), nor that QM results are predetermined by some hidden reality, or hidden variables.