Is there as much to gain as what has been achieved with electric fans?
Remember you aren't gaining anything. Just like a belt run fan, each is a parasitic loss of power. When you put an electric water pump on the engine, you are regaining some of the lost power which you didn't have access to in the first place. This will allow for better fuel mileage, etc. You can expect about the same difference as what you'd see from an electric fan, but every engine is different so it would be very hard to put a specific number or percentage associated with the change.
Should the water pump start spinning when the thermostat opens?
No. It should run just like a regular pump does, which is all the time. I think you may have a misconception of how any pump works or even how the cooling system operates. There are two sides to the cooling system which is separated by the thermostat. What I'll call the inside includes the engine and water pump. The outside includes the radiator. The heater core can fall either the inside or the outside depending on how the manufacturer set it up when engineered. On the inside portion, the coolant circulates constantly. On the outside, the coolant flows only when the thermostat is open. Without the circulation on the inside, you'd get hotspots which would produce steam (these still occur even with circulation, but that's another story). The circulation helps keep the engine at a near constant temperature which helps with efficiency and with longevity of the engine.
Could the thermostat be removed and the shutting down of the pump be relied upon to stop water being cooled by the radiator when the engine has not come up to its recommended operating temp?
No. I believe this is answered above.
Must the proposed electric water pump always spin albeit at some slow speed to keep the engine temp even?
An electric pump flows what it flows. It turns on and circulates the coolant at the speed which it's rated for. This circulation speed is usually much faster than regular water pump at idle and a bit slower at higher engine speeds. They are usually designed to flow what the engine needs at any given speed.
Would variable speed be good or would a simple on/off thermostatic switch suffice?
Neither. You want to keep the flow at an even pace. You could possibly create some type of rheostat which would adjust speed of the pump to coincide with engine speed, but why would you when they work perfectly fine without it. By adding something like this you are creating unneeded complexity. With complexity you're adding more chinks in the armor ... one more place for something to fail.
Should the pump be allowed to run after the ign is off to facilitate cool down like some electric fans do when the engine is hot?
While most electric pumps don't come this way, it can be rigged as such. It's actually not a bad idea, especially if you have water cooled turbos or other parts which can get very hot and are cooled by the coolant system. Hot spots anywhere in the engine can cause issues. By having coolant flow over them you are helping them to cool at an even rate, which will allow for them to last longer and perform better in the long run.