My words literally betray me.
Must a player make exactly four moves, or can less be made, or none?
Yes, but this is rarely desirable.
Once a gate is sealed, can a new cultist or spawn appear on it during the summoning phase?
Yes. But you can't use it for travel and shuggoths won't move towards it.
Also, I'm unclear what flipping a sealed gate to the elder side function is.
As the card says, nothing spawns in the town that is sealed.