Translation:The sister is in the city, but the brother is at home.
Note: urbe is the ablative form of urbs (3rd declension) meaning city. In Latin, nouns are declined for case - different declensions of nouns are declined in different ways. When the ablative case is used after the preposition 'in', the resulting meaning is to be in somewhere. Domi is the locative form of domus (2nd declension). The locative form is a very rare form in Latin, and only a few words, such as domus, employ it. It has the same meaning as the aforementioned ablative construction, namely 'in/at'.
In this case, I think that's wrong. "Sister is in the city..." in English is short for "My sister is in the city...", so I'd have expected something like "Soror mea in urbe (est)..." in the Latin if that's what they'd meant.
More generally, yes, you'll often have to add "the" or "a(n)". Latin is a synthetic language (fewer words in a sentence, but more forms of each word) while English is an analytic language (fewer word forms, but more words in a sentence), so you'll almost always have to add more words in the English translation to capture the sense of the Latin. That will be the case especially when they course gets extended to the more-advanced uses of the dative and ablative (you're already seeing some examples, e.g. when you have to translate "Romae" as "in Rome").
No, I disagree. The use of "sister" without an article does not necessarily imply that she is "my" sister. "Sister is in the city" could refer to a woman who is a member of a religious community. Rather common in a Catholic context. The same with Brother. And the usage of "sister" and "brother" in the African-American community is somewhat broader and different than in Standard English. It's all a mater of context. And we have no context here. Every possible translation, therefor, should be allowed. "Sister is in the city but brother is at home" should be allowed because in some contexts it is the correct translation.
And further, consider the following; Mater scribit et frater domi dormit. This exercise allows, "Mother writes and brother sleeps at home," as a correct translation. No a, the or my.
I don't think so, because, if you have a text, talking about the sister of someone, for instance:
John and his sister are at home. They sleep. Someone knocks the door.
The sister get out of her bed in order to watch though the window to see who's knocking the door.
That's true that the possessives are often implied in Latin, but in this sentence, it's difficult to say if the narrator talks about his or her own sister of the sister of the person he addresses, or whoever: no context.
I think it's more obvious when you havea structure like I.... the parents = I visit my parents.