"And your dad?"
Translation:Und dein Papa?
Shouldn't "your father" be in the accusative case, i.e., "deinen Vater?" Well, obviously not, but I don't quite see why. How about the phrase "your ball"? Wouldn't the ball be the direct object that one possesses? Are these examples nominative because there is no verb describing how the subject of the sentence affects the direct object?
Pretty much, yes -- without context, the default case is nominative.
Another case would be used if there is an implied verb, e.g. if this is in response to something.
"My dad is a doctor. And your dad?" -- implied: "is your dad a doctor?"; "your dad" is the subject so it would be in the nominative case.
"Your mother loves you very much. And your dad?" -- implied: "does your mother love your dad?"; "your dad" is the direct object so it would be in the accusative case.
"How are you? And your dad?" -- when using es geht ... (gut), the experiencer is in the dative case, so it would be Wie geht es dir? Und deinem Papa? with the implied verb being Wie geht es deinem Papa?
It could be 'Und dein Papa/Vater' but 'Vater' is more like 'father'. But what's in plural here?
Dein doesn't mean singular. Der/das words use dein Die words use deine Vater = father Papa = dad/daddy