If you use a electric water pump for drag racing should you use a thermostat? What about on the street?

Always use a thermostat to allow the engine to operate at it's required temp.

Electric pumps don't move enough water for street use.

There are many guys successfully running electric water pumps on the street. One advantage of the electric pump is the constant flow rate so your low speed cooling will be more efficient. Meziere makes units up to 55 gpm, which should take care of any street application.

For what it's worth...
My mezzeire does not have a bypass port.
My intake manifold is partitioned in the water outlet and does not accept a thermostat

I have been running the 55 GPM meziere on my street car for 2 years. Works great. Also running Spal fan and 180 thermostat. The car never runs above 190.