Three questions of ambiguous nature asking for "Yes or no" answers

1. Whenever I see sentences like the one below, I tend to feel uneasy in part because, I think, I am too stuck on the idea that a plural subject would be likely to be followed by a plural object or an object of a preposition. Are they OK? Are they basically saying the same thing?

How many students have an apple on his/her desk?

How many students have apples on their desks?

How many students have apples on their desk?

How many students have an apple on their desk?

How many students have an apple on their desks?

2. Would you say the writer's placement of the article "the" is OK here? the is not a part of the proper name, I think.

Would you like to come to the ZZZ Center's anniversary party?