When I put the key in the outside cylinder it is free spinning without locking the lock.
The tailpiece on the outside cylinder has been cut to short or is not connected to the hub.

The lock will not deadlock or function properly.
The tailpiece on the outside cylinder has been cut to long as is protruding into and interfering with the lock mechanism.

The safety release on the 001 deadlatch is not operating properly.
If the lock has been re-keyed, the cam on the back of the inside cylinder (which operates the safety release) has not been reset correctly the safety release feature will not work.

I have an outward opening door, can the 001 deadlatch be fitted in this situation.
Yes, but if the door is outward opening or metal frame you will need to order the version of the 001 to suit these applications.

I lock the 355 deadlock from outside with the key, but the inside does not deadlock.
The deadlocking of the 355 is a very manual process, as the internal and external locking operate independently.