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Yes, but only if you're talking about more than one cat.
Mag ihre Katze Milch? = Does her/their cat like milk?
Mögen ihre Katzen Milch? = Do her/their cats like milk?
By the way, please don't ignore umlauts. In some cases umlauts change the meaning of a word and they're always pronounced differently. To a native speaker, reading texts without umlauts is like nails down a chalkboard. If typing German letters is inconvenient for you, you can use "ae" instead "ä", "oe" instead of "ö", "ue" instead of "ü" and "ss" instead of "ß". That's what native speakers do when they're using a non-German keyboard. Mind you, this is only a makeshift solution. If you Google "how to type German letters", you'll find lots of great tutorials that show you how to type them properly.
The word does appears only in English -- most other languages don't have a corresponding word. A phrase like "Er isst" is translated three possible ways in English: He eats, He is eating, and He does eat. When it's a question "Isst er?", in English you almost always have to add the "does", even though there's no corresponding German word.
This question was already asked. No, that translation can never work. We have the word "Ihre" here, and the first thing to notice is it's not at the beginning of the sentence, yet it is capitalized. That tells you it's using the capitalization convention of the "formal you", and we can end the discussion right there without even referring to pronoun grammar tables. Even if you didn't know what "ihre" meant, the only noun/pronoun that get capitalized, no matter where it sits, is the formal you. Now, when it comes to pronouns, there is a potential point of confusion and you probably stumbled on it. Consider these three sentences: a) I see her cat. (possessive pronoun) b) I see her. (direct object pronoun) c) Give the cat to her. (indirect object pronoun) Same word "her" in English gets rendered in German as:: a = ihre b = sie c = ihr depending on usage I put in parenthesis. In order to get good at German pronouns, you're going to need to memorize quite a number of tables. (For sure: Subject, Direct Object, Indirect Object, Possessive, Reflexive, Demonstrative, and I might be forgetting some!)