You do need an expansion tank. It should go in the cold water line between the cold water shutoff and the tank inlet. It provides an air cushion. Each tankful of water will expand a little when heated and can trip the temperature/pressure relief valve if there is no expansion tank.
(The expansion tank will work anywhere provided there is no closed valve between it and the water heater but the inspector likes to see it quickly when he looks at the water heater.)
I have not read about drip tees in flues but I have heard about water vapor condensing in the flue and running back down into the furnace or water heater.
The good conscientious technician or serviceperson will carry extra oils and lubricants in case the new pump did not come with oil or the oil was accidentally spilled, so the service call can be completed without an extra visit.