Translation:The woman writes, but the man does not write.
I don't know if 'non' is allowed to stand by itself in Latin. Because Latin verbs encode person as well as the action and tense, it may be required where it wouldn't be in English (The woman is writing but the man is not).
Someone who's more well versed in Latin will have to confirm, but I'd go with the pattern being taught for now.
Some of this vocabulary was imported directly into Esperanto, like 'sed' which is unchanged (if you say 'sed' in Spanish, it means 'thirst'). 'Vir' is another word, but an '-o' was added: 'viro'. If you want to say 'woman', just add '-ino'. Esperanto doesn't have gender, so 'la' is used for 'the' throughout. Here's the sentence in Esperanto: "La virino skribas, sed la viro ne skribas."
Exactly. When you translate from one language to another you don't translate each word as you see it; you translate the meaning.
Mae'r ddynes yn sgwennu ond dydy'r dyn ddim yn sgwennu would be the Welsh translation, but word-for-word it's "is the woman -ing write but is not the man not -ing write" which is obviously not an acceptable English translation. The translation is "The woman is writing but the man is not writing".
"Feminae scribunt, sed viri diurnalis non scribit" will be an example when you could interpret the sentences as being spoken in indefinite article, but in all ways it could be also be interpreted as a definite article. The problem with Duolingo is you don't get natural conversation in form of stories in Latin, but luckily you can get help in many free websites that teach the latin language with a quick google search or just looking at the latin forum in Duo