Can the sciences legitimately distinguish between true and false religious beliefs? Why or why not?
No.
There is no definition of "God", "supernatural being", "supernatural", "spiritual", "immaterial" &c useful to reasoned enquiry. This results in questions regarding such things being not so much unfalsiable as meaningless.
Is there anything about the scientific method or methods of inquiry that would prevent the sciences from legitimately distinguishing a true religious belief from a false one?
If the question can put in falsifiable form, then yes.
Can the sciences legitimately tell us whether the notion that Rama's Bridge was built by the Vanara army of Rama is true or false? Why or why not?
The geological evidence points to its formation as a natural occurrence. Further, since Rama is a supernatural being, the question of Rama's hand in its construction is meaningless.
Can the sciences legitimately tell us whether the notion that there was once a great flood covering all the earth is true or false? Why or why not?
Yes. No evidence suggests that the entire earth was submerged at any time in earth's 4.5 bn year history, and certainly not at any time in the two million years since genus Homo first emerged.
The evidence against a Noah's flood in Ussher's timeline, which places it in 2348 BCE, is overwhelming. Such an event would leave a single flood stratum all over all continents and islands and the ocean floor. Geologists find no such stratum. Such an event would create a genetic bottleneck in every species of land animal, all the bottlenecks having a common date, and there are no such bottlenecks. Such an event would require some 1.113 bn cubic miles of water over and above the water on earth today. It's not here.
And there are countless further reasons, not least the uninterrupted existence of civilizations from that era on both sides of the purported flood, in Egypt, Mesopotamia, the Indus valley and China.
Can the sciences legitimately tell us whether any specific deity exists or doesn't exist? Why or why not?
Why not? Because the category 'deity, a necessary aspect of the target of the enquiry, has no coherent definition.
Can the sciences legitimately tell us whether enlightenment is an actual state of awareness? Why or why not?
I don't know what exact state of mind 'enlightenment' is intended to designate. If it has a definition useful to reasoned enquiry then the question can in principle be examined and a result obtained.
1) Assuming that the sciences could indeed tell us whether at least some religious beliefs were true or false, then in what way(s), if any, would it matter that the sciences could do so?
Assuming the problem of meaningful definitions is implicitly overcome, then beliefs about the good life, morality, the obligations and rights of citizens, the afterlife, judgment, heaven, hell and purgatory, or reincarnation and transmigration, could be answered on an objective basis.
But since these are largely questions of detail within known parameters, in reality the humans involved would need to agree on a very clear statement of objectives and priorities before any such assortment of opinions could be winnowed.
2) What, if anything, is the relationship between scientifically established fact (and/or hypotheses) and the meanings or purposes of religions?
That appears to be a question for anthropology. So far anthropology has been longer on hypotheses than on firm conclusions. So the answer to this question is, in reality, not a little muddy.