So I arrive at 10pm at Beauvais Airport and I found that trains and shuttles from this airport to Paris are really expensive, but I found a fairly cheap ride out early the next morning, so I'm curious if there are any rules that say I can't stay at the airport overnight even though I have no flights connecting flights from there and really no reason to stay in the airport. Any advice?
Aéroport Paris Beauvais Tillé airport closes overnight so, unless you're prepared to doss down out of doors somewhere (uncomfortable as well as risky) I'm afraid you'll have to sort out a 'plan B':
They are probably closing the airport and you will be asked to stay outside. It is not possible to stay inside during the night. I would suggest taking the bus for around 1eu to Beauvais town center and finding someone to sleep at during the night.