Can explorers choose not to move? So instead of rolling dice or playing an action can a survivor skip their own turn so they don't leave a room? One example would be to stay in the mystic elevator and then each turn use the mystic elevator to travel around the house.
Yes, you can choose not to move. The rules on movement state:
On your explorer’s turn, you can move up to number of spaces equal to your character’s current Speed.
"Up to" includes 0 spaces.
Note that if you stay in the elevator for your entire turn, then the elevator will move again. From the FAQ:
Heroes must roll for their destination floor each time they enter the elevator, or at the end of each entire movement phase that they spend in the elevator without moving.