Minimum use height (in any autopilot mode) is 140ft for 737-700/900/900ER, 138ft for 737-600 and 158ft for 737-800. That's for aircraft with EASA AFM, it might vary for other authorities.
Boeing recommends manual flight = manual thrust for all of their non-FBW models, except in climb (where thrust is fixed). I believe for FBW this isn't the case and A/T can be kept on.
Autothrottle use is recommended during takeoff and climb in either automatic or manual flight. During all other phases of flight, autothrottle use is recommended only when the autopilot is engaged in CMD.