Is it dangerous to leave a hose spigot on with a closed nozzle at the end?
As long as the attached timer is shut off and no water is flowing the pressure in the water line will be no different than what it would be when the spigot is turned off. 

You water bill will reflect how much water that you actually use. The only way it could go up astronomically would be if the timer broke off the spigot or failed in an open mode where water was allowed to flow freely from the spigot for long periods of time.