My sliding glass door to the patio, after your lock it and check it, moves a little bit to the left. It is still locked, but it moves. Is this normal and safe?

Yes, the lock is a little latch that comes down, it has a curved section, and it's just the curved section pushing the door out so that the latch is more securely in place.

As long as there isn't a gap I would say this is perfectly normal.

I find doors that have slack are easier to jimmy open. Take up slack, and/or install additional lock.