HDMI splitter will be your best bet I think.
The projectors and resolume will work, but you will need a mini(applenonsence) to VGA adapter, then an active split.
Avoid passive splits if you are sending the data any distance. Active splits are definitely a better option, even if it means they go on eBay afterwords.
It is possible to send video over ethernet, but i wouldn't bother.
Not for a quiz.