"My daughters sleep in the city."
Translation:Filiae meae in urbe dormiunt.
Are you including prose examples when you say "a lot"? Vergil often leaves out the prepositions in his poetry, but in classical prose authors like Cicero and Caesar, the preposition is used with the ablative. Certainly on standardized tests of classical Latin grammar, the preposition is expected when expressing place where (excepting locative case uses, of course).
The Ablative is also used, when you want to say "where" something happens (it has quite a lot of usages from tellint you the price of something to with whom you are doing something). And also if you're moving away from something as you stated correctly (often accompanied by "e" or "ex", but not necessarily.
mei is also the genitive case for ego. JuanaGoGo is making use of the genitive to represent the 'possession'.
EDIT: Seems to be that the possessive adjective for pronouns is preferred over the genitive (Potentially incorrect to use the genitive? Or maybe just less common? Can someone verify that?).
In the Duolingo sentence they are making use of the first person singular possessive adjective meus, mea, meum (and it matches the case and number of filiae [nominative plural]). This is what Duolingo mostly uses.
e.g. liber meus -> 'my book' | libri mei -> 'my books'
The dative of possession makes use of the possessor in the dative case and usually the verb sum, esse ('to be'). You have (most likely) seen mihi nomen est, "my name is".
e.g. est mihi liber -> "I have a book" | est mihi libri -> "I have books"
The genitive of possession puts the possessor in the genitive case (before or after the noun it modifies). There are other uses of the genitive which do not specify possession. From the provided link it seems like for pronouns (like ego) that the adjective is used instead.
e.g. liber Ciceronis -> "Cicero's book"/"the book of Cicero" | libri Ciceronis -> "Cicero's books"/"the books of Cicero"
The dative of possession requires esse while the other do not. When the adjective or genitive is used with esse it emphasizes the possessor while the dative emphasizes the possession.