Would there be a problem installing 20A switches on a 15A circuit?
You are correct on both counts: 


There is no problem with using a 20A switch on a 15A circuit, as the breaker will trip and protect the switch from over-current. 

Using a 15A switch on a 20A circuit is potentially dangerous though: the circuit may draw up to 20A, but the switch will be overloaded at that point which may cause it to overheat/melt/start a fire. 
Really this is the same as wiring: you can use larger wires (with higher current ratings) but not smaller wires

A 20A outlet is not allowed on a 15A circuit. It's not really "dangerous" since the breaker will (should) trip at 15A anyway, but it is bad to knowingly install it this way.