Firstly, does running the engine at colder temps affect HP? (bad, good or other)
Depends on which part of the engine is hot or cold.
High temperatures increase the thermal efficiency of the engine, better fuel vaporization, less loss of combustion heat to the cooling system cylinder walls.
High temperatures increase likelihood of detonation, and reduce the volumetric efficiency of the engine (lower density intake air charge).
High temperatures in the intake increase the speed of sound in the intake air, which slightly delays onset of choke flow in the intake ports, since that is tied directly to the speed of sound in the intake mixture.
High temperatures lower viscosity of the engine oil reducing pumping losses in the oil system, but it also increases friction due to thermal expansion of parts (piston side clearance drops as the piston gets hotter).
In short it depends on which effect you want to maximize and if the engine can tolerate the change. A rule of thumb developed by NACA during research on high performance aircraft engines is that power increases at the square root of the change in intake air absolute temperature. Cooler intake air charges producing more power than hot intake air charges, but you are in a running battle with friction losses and pumping losses of both intake air and oil windage, so to a point high temperatures reduce those losses. Cold engines are much harder to crank than hot engines due to thick oil and as the oil heats up power losses go down until you reach an ideal temperature then begin to go back up.
Ideal oil temps seem to be around 210 Deg F, and ideal coolant temps are typically near 180-200 degrees but each engine has its own sweet spot for both values.
I would not want to completely stop pumping coolant to avoid development of hot spots in the head and local boiling of the coolant.
Your idea has merit if you set it up with double water pumps and if the engine gets too cool one of the pumps shuts off to keep from over cooling it. If I were you I would run some sort of thermostat, perhaps only a 160 deg F but it will help even out the temperature. It also helps the water pumps develop hydrostatic pressure in the cooling system to help suppress boiling, since the pumps are moving the water through a resistance rather than wide open with no back pressure. Too low of a pressure at the pump can also lead to cavitation.
Years ago I found my 392 Chrysler Hemi gave me best times at the drags if I idled it just long enough that it was difficult to hold my hand against the front of the cylinder head prior to staging. This gave a water temp at staging of about 140-150 degrees. By the time I was staged the water temp was around 170 and I would trap at 180-190 deg F. That worked for me and that particular engine. You need to find what temps your engine wants to run at with the oil you use and your running clearances.
(something to test on a dyno)
Larry