I learned that back in HS with Spanish. The variations drive me nuts with German. Der, dem, den, die, das, des. English may have a billion synonyms from stealing from every language, but we don't assign genders to everything. Why is a table masculine in German, but feminine in Spanish? It's a table.
There are three translation of English you in German:
The personal pronouns (here in nominative) are:
1) du (= informal, sing.)
2) ihr (= informal, plur.)
3) Sie (= polite form, sing. + plur.)
They correspond with the respective possessive pronouns (here in nominative, masc./ fem. / neut.):
1) dein Vater, deine Mutter, dein Haus
2) euer Vater, eu(e)re Mutter, euer Haus
3) Ihr Vater, Ihre Mutter, Ihr Haus
In the sentence above, the cat is in dative case in German, and the possessive pronoun needs to be in the corresponding form as well:
2) eu(e)rem Vater, eu(e)rer Mutter, eu(e)rem Haus
In spoken language, you will often hear another variant for dative masc. / neut.:
euerm Vater, euerm Haus
Even when listening to the slower track, the second r in "eurer" sounded like it was dropped off. It's been ages since I learned dative case, and I didn't even learn it very well when I did, so I had to rely entirely on the audio when writing this and it sounded exactly like "Wie geht es eure Katze?" to me
The question "wie geht es..." literally translates to "how goes it...", and then it's the dative person/thing: wie geht es dir (you), es geht mir gut (me), wie geht es ihr (her), etc. So 'es' is in the nominative case, as 'it' goes. This phrase, while super high frequency in the language, isn't necessarily super easy to decode grammatically, so hopefully this explanation has helped.
The way I keep this straight in my head is to assume this expression is similar to "mi piace" which is used to mean "I like it", but translates literally to "it is pleasing to me". Some expresions are simply backwards (English has the subject doing the action and German has the action being done to the subject) and "___ geht es" happens to be one of these.
Yes, translating into Southern English, this would translate into:
How is y'all's cat?
Or in direct translation it's:
How goes it (for) y'all's cat?
Katze is feminine so it uses "Die" but because we're talking about how it goes for their cat, we change "Die" into the genitive "D(er)" which corresponds with "euer". :)
Wie geht's euer Katze?