That's the accusative case marker. The woman is the direct object of "see" (she's the thing being seen) and nő is in the accusative case as nőt to mark that.

A bit like in English we would say "I see him" and not "I see he" -- the "him" has to be in the objective case as the direct object.

No. "look" and "see" are different words in English, just as "néz" and "lát" in Hungarian.

Also not "the woman".

Yes, either is possible. You can also go full circle and add "Egy nőt lát a férfi." And a couple more.

• A férfi lát egy nőt. - Pretty neutral. If you put a bit vocal emphasis on "a férfi", it gets focus on the fact that the man is doing the seeing, not anybody else.
• Egy nőt a férfi lát. - Same as above, but a bit more uncommon, since the topic here, "a woman", is indefinite. You can use it if the topic of the talk has been about women in general.
• A férfi egy nőt lát. / Egy nőt lát a férfi. - Here the focus is on the woman. That's the important part here, that's what the man is seeing.
• Lát a férfi egy nőt. / Lát egy nőt a férfi. - A more uncommon variant. The focus here in on the action itself: seeing is what the man is doing with the woman. He's not talking to her or anything else.

A better example where you could use the latter word order might be the following dialogue:

• Mit csinálnak? - What are they doing?
• Nézi a férfi a nőt. - The man is looking at the woman.
• Nézi? Sem beszél sem érinti meg? - He's looking at her? Not talking or touching?

It sounds pretty clear to me. Maybe a bit short, but that's pretty much the fate of any article in any language. :)

Hm, it should be alright. Feel free to report it.

Usually "woman" is translated as nő, and "lady" as hölgy, but I feel like the levels of politeness for using each are different between English and Hungarian. You'd use hölgy mostly for older women and/or women of a high social status, while "lady" has a broader use.