So I have a <div>. I would like to turn the css style overflow:visible on all content that pass through the top of the <div> and turn overflow:hidden on all content that is below the div. Is this possible?

Why don't you wrap all that is below your <div> in another <div> and make this last one overflow:hidden.
Does this answer your question?
Good luck!
EDIT:
Do you meen something like this example?