Relays: my use for truck relays is switching on heaters in my thermal storage water tank. Not big ones, though - I use relays rated for 24V and 40A of current. Since they are old, I have applied a safety margin and only let 25 A flow through them, so each of them handles 24 x 25 = 600 W.
As for using DC appliances: benefits do exist. If a household has a low voltage DC battery bank (some do, some don’t) then dropping the battery voltage a few times to power car parts comes with a smaller efficiency loss. In my household, DC appliances are used for lighting, communications, computing, cooling food, pumping water and soldering electronics. The rest goes via AC. I think a car air conditioner could cool some small storage room decently. With big living rooms, it would have difficulty since it’s a small device.
On individual scale, precisely that - a split type AC with one half indoors (or in a water tank) and the other half in an outdoor environement (air, water or ground).
If you’re extracting heat from the environment, the machine lets the working fluid evaporate into the outdoor heat exchanger and compresses it back into the indoor heat exchanger. If you’re cooling your premises - reverse that.
However, on a city scale, it’s like “you’ve got a lot of sewage at 30 C” -> “your heat pump is a large building” -> “your sewage outflow is now at 10 C, but your underground heat reservoir gets charged to 140 C (stays liquid because of water column pressure), and you spend much less energy pumping the heat than you would spend heating the water directly”.