No reason to run CO2 24/7. Or use an airstone. Pick up a $5 analog timer, plug the solenoid and lights into it, set it for 8 hours a day or so, and done.
If you want to play with it, you can get two timers, and run the solenoid an hour or so ahead of lighting, as many do, but, honestly, there's no confirmation of this having any effect whatsoever on growth beyond what having the lighting and CO2 on and off at the same time does. Besides, with a paintball can, you want to conserve CO2 as much as you can. As it is, it's 1/4 the size of a 5# cylinder. Without a solenoid, you'll use 12 paintball cans for each 5# with a solenoid.
A 3-way solenoid act identically to a 2-way solenoid if you A) plug the 3rd port (if there is one) or B) run a check valve anywhere beyond the solenoid. But keep in mind that if there's an "M" in the model number of the Mouse, then it requires a mount, available only through Clippard.