It's even worse, it also behaves like an adjective. ...Okay, that doesn't really make a difference, save for using -ak over -ok for the plural.
To answer your question: yes, it's an odd one. You will find some words that have a stray é in them but otherwise follow back vowel harmony. For instance acél (steel, acélja - his steel) or béka (frog, a békának - for the frog). There seems to have been a different back vowel sound in olden times that has been since replaced by é.
There are two nouns, that I know of, that have strictly front vowels only but still attract back-vowel suffixes, both of them with the vowel é: cél (célok, célom, célod) = "target, goal" & héj (héjak, héjam, héjad) = "shell, peel (of a fruit)"
The explanation is probably a vowel shift from the close vowel í... or a back vowel.
The English here is grammatical. But I think most English speakers in most cases would say 'the men and women' rather than the 'women and men' -- perhaps out of custom. Anyway, my question is: Does Hungarian etiquette (or perhaps euphony) favor mentioning the women first, or doesn't it matter much?
The order in English is almost invariably: men, women, children. "The women and men" is unnatural - one could say "the women and the men" if for example there were a lot of women and only a few men. It may be politically bad taste but if you want to put women first, you might say: the women and (their) husbands