My notes show this verb takes Accusative. This is Genitive, correct?
You are correct x2. Yes, this verb takes Accusative. And therefore it takes Genitive after it gets negated.
Why was it 'jabłek' in this sentence but in 'nie lubię starych jabłkach' it was jabłkach? I thought they were both genitive
Jabłkach is locative. You probably got confused by the fact that for adjectives, genitive plural and locative plural look the same.
I don't think so. "Nie lubię starych jabłkach" is not a correct sentence.
It should be "nie lubię starych jabłek" as well. Are you sure you remembered that one correctly?