Electronic – Good board for DIY temperature control/readout

1000 yards when you don't want to run a cable is tricky.

Like you said, that's too far for reliable WiFi. You can probably boost WiFi with a directional antenna on each end. That wouldn't be legal, but if you're careful that the directional beams don't spill off your property then in practise nobody is going to care.

You could rig up something using a 802.15 radio module in the 9xx MHz band to periodically send temperature readings one-way. Normal 802.15 radios won't go that far, so you'd make your own directional antenna at the dugout. A dipole with a parabolic reflector behind it would probably work. You do the same thing at the receiving end at the house. This won't be legal either, but you'd be transmitting one-way from the dugout up the hill only. If yours is the only house on the hill, then nobody else would even notice and would have no reason to care.

Another option is injecting a modulated carrier onto the AC line. There are "X-10" modules for this, but I've heard mixed reviews. There is a lot of noise on the AC line, making it difficult to use it for communication.

A better answer might be to spend the money instead on insulation. It sounds like this pump house is partly dug into the ground. By adding a few inches of ridgid foam insulation on the walls that go to the outside air, you can probably keep the temperature above freezing all winter. I have a well pump at the bottom of a shaft that is maybe 6 feet deep and 3 feet in diameter. It only has a concrete cap on top, but the water in the pump doesn't freeze even during cold winter nights.

Yes, a process controller with 10Hz or better sample rate and analog out should be fine with this. Don't try to use the el cheapo ones.