I don't think the second article before "man" is required for a correct translation. "The/a woman and man are eating" is also correct (I'm a native English speaker and this seems fine to me).
Both are completely valid. Because in polish there are no articles ("a" or "the") the translation is a valid one. Your translation is also correct, so there is no error here :)
How do you say, "Man and woman eat" in Polish, if not with this sentence?
Eats won't do?
No, "eats" is the third person singular form, and you need a plural verb for subjects joined by "and".
I cannot hear the difference in the sound when words end in a, y, a-hook, or i. Is there a resource where I could listen and compare all of these at one time?
Maybe you can try this: http://popolskupopolsce.edu.pl/lekcja/1/MAZOWIECKIE
(I would disregard "ch", it really should be identical to "h" in standard Polish)