Seems kind of unfair to suddenly expect us to translate names after having not done so for the last 95% of the course...
"Guess what Zsuzsa is going to be called! Will it be Zsuzsa? Susannah? With an 'h' or not? Susan? Susie? Sue? Welcome to... WHEEL! OF! ZSUZSA!"
why is it katiNAK instead of katiNEK? Is there no vowel harmony with possession or with names?
It's pretty simple here, the letters 'i' and 'í' are neutral and generally follow the harmony of the rest of the word. Kati is a back-vowel name.
In my observation, when names have mixed vowels, the harmony in Hungarian tends to be with the vowel of the first syllable (thus kAtinAk) rather than the vowel of the last syllable, the way Turkish tends to do it.
I don't know whether this is a reliable rule, though.
Last syllable for compound or foreign words; mixed vowel words which are not compound or foreign words take a back vowel.