Both "Gen Z and their…" and "Gen Z and its…" are fine. I personally prefer to treat a group as a plural.

"The class tossed their graduation caps in the air" strikes me as less clumsy than its alternatives.

Well, considering Gen Z is a group of people who has parents, their works just fine. "my Mom" "my Dad" ... parents are usually referred to as entities the child has. And Gen Z is a plurality of people.

No, it shouldn't. It's perfectly valid to refer to the members of a group rather than the group itself.