NO COLD WATER
"Solenoid". No, you probably don't need one. You shouldn't even HAVE one.
I suspect strongly that you need to prime the new pump. There should be a priming plug on the top of the pump. Remove the plug and pour water into it. Replace the plug quickly and turn on the pump immediately.
You may need to do this several times before the pump starts drawing water. You can hear the change in pump sound when it starts moving water.
It does help, while you're priming, to close any valve you find between the pump and the pressure tank. As soon as the pump draws water, slightly open that valve or the pump will cycle on & off very rapidly. When the pressure tank is pretty full of water (you're getting fairly significant pressure on the gauge), open that valve fully.
I always liked to have a globe valve on top of the pump with a bell reducer and cap above that for priming purposes. The globe valve permits VERY fast pump repriming, and the bell reducer acts like a funnel, making it easy to pour water in. With the globe valve closed (normal operation), the cap doesn't even have to be tight - I need no tools to prime my pump that way.