Do hotels have the ability to kick out people?

Sure, they are allowed to cancel any reservation that they do not want to accept. Even if the room was pre-reserved, someone has to be held responsible for the room. If the occupants show up and are not the correct ages (many hotels require an occupant to be able to show ID that says they are 21+) or do not have a major credit card to be kept on file, then the hotel has every right to not honor the reservation. If the hotel takes a look at your group and just decides it doesn't want you there, based on your ages only, they are totally within their rights to do so. They should return the deposit, but they don't have to give you a room. Conversely, if you show up and are underage, then the hotel should not have to refund your reservation fee, after all, you committed fraud when the reservation was made, even if you didn't read all of the fine print.

ETA- Not saying fraud in that you stole the credit card, but fraud in that you made the reservation stating one thing (even if you weren't aware of it) about minimum ages of the occupants, and then kids show up who aren't that age. The hotel can turn you away on the spot, refuse to rent you the room, and keep the deposit.

In that situation no. However, if the graduates were breaking hotel rules then yes they can kick people out. (ex: making very loud noise, pushing other hotel guests in the pool, pulling the fire alarm aka things on did on my senior field trip that got us ALL sent home and suspended)