Learn German: Wie hat sie die Katze genannt?

It comes from the verb nennen (to name/call). It is a mixed verb, meaning it mixes rules from regular and irregular verbs. Like regular verbs, it adds ge- at the beginning and -t at the end. But like irregular verbs, its stem vowel changes. Here, the e in nennen changes to an a to make genannt.

Why does the sentence start with Wie (how) if the English translation says "What"?

In German, you use wie (how) when asking about names. You might remember this from the basic phrase Wie heißt du? (What is your name? / literally: How are you called?). The same logic applies here: Germans literally ask "How did she name the cat?".

Why is genannt at the very end of the sentence?

This is the standard word order for the Perfekt (spoken past) tense. The helping verb hat takes the second position in the sentence, right after the question word Wie, while the past participle genannt is pushed all the way to the very end.

Is die Katze the subject or the object here?