I see a lot of sentences in these practices that put the feminine noun first in compound subjects. You see this is some stock phrases in English - "ladies and gentlemen". The rules of grammar do not require this but convention does - if you don't do it then it doesn't sound "right" to a native speaker. Is there such a convention in Korean? (Or is this a bias of the writer of the software?)
I don't think so. I'm not a fluent speaker yet, but having consumed some Korean media at this point, I'd say it varies. Just as one example, at the live final of Mixnine (a competition show), they kept asking whether it would be the boys or girls who won - in that order.
For that matter, the phrase "boys and girls" sounds more natural to me than "girls and boys" - so I think it's variable in English, too. "Kings and queens" also sounds more natural to me than "queens and kings," for whatever reason.
Okay, I'm still not very advanced, so I'm not 100% confidence in this answer. It's worth a shot, though, so here's my best guess: I think that "같이" roughly means "together." "함께," however, means something closer to "with ___." I was going to give you an example of a phrase I learned (the Korean version of the phrase), but I'm afraid my grammar will be off (still working on that), so you can just have the English translation of a phrase: "I eat bananas with coffee." (Don't ask, it was from Duolingo, after all.) And in that example, they used the word "함께." (to mean "with.") I do believe "함께" can also mean "together," though. So, in conclusion, I think "같이" means "together" and "함께" just means "with," but still could work as "together." (Again, I'm not 100% sure if this is correct. One last thing: I tend to make replies longer than they need to be, so I'm sorry for that... Congratulations if you read that all. I hope this was at least somewhat helpful...)
I'm not sure if this applies to every situation, but here's my guess: (Just a reminder: "여자" means "woman" and "와" is pretty much a particle for "and." "남자가" just means "man.")(not sure about the particle, I'm still working on that) Okay, sorry for that, you probably already knew that. Because the word woman came before man, I think that's why it'd be in that order. However, it wouldn't change the meaning of the sentence if you swapped around the two (woman and man) anyways... I'll just leave it at that before I ramble on for another paragraph.
Generally if the -는/은 particle is used, it's more likely that 'The' would be correct, but when 이/가 particle is used it could be either. The point is that the definite/indefinite distinction simply isn't made like that in Korean so YOU need to pick which English word to use based on context. And sometimes either could work (like this question). And often on DL, you have no context. So don't fret too much about this and if you think DL unfairly marked you as wrong, report it.