Yes, but wasn’t she the one who brought and handed it to him?
I would have to assume so. There is no way it was a 'surprise' sprung by Fallon. She would never allow it on air if it had been.
She was trying to make fun of the issue to downplay it. She was trying to show she has a sense of humor.
I would again assume this was created/originated by her staff after a focus pull where they were desperate to find a way to shrug this off.