Why won't the door with the Turning thing open?

Which door are you referring to? There is a handle that you place in the basement and turning it will then open the cage around the front door.

If you're playing on extra lock mode you won't be able to use the spin wheel to remove the cage from the front door until after you've found the wrench and unscrew the bolts that are now attached to the cage. I was wondering this myself when I first tried extra locks for I didn't notice the small hinges at the bottom the cage that weren't there on the previous play thru.

The door with the hand wheel is not a door. It opens up a cage on the front door. I had this problem, too. I kept trying to push the door open.