

No, your UPS is tailored to the characteristics of the lead-acid charge/discharge cycle. Some LiFePO4 battery assemblies are built with charging electronics that simulate lead-acid for a drop-in replacement, but those are never cheaper than traditional lead-acid batteries.
Zigbee or Zwave temperature/humidity sensors are common. Add a 3-circuit relay box and you can simulate the behavior of pretty much any thermostat with a few rules.
HA or any other system that can toggle outputs based on sensor thresholds would work just fine.
there are some subtleties with real HVAC thermostats, like running your AC compressor at least five minutes and ensuring that it stays off for at least 5 minutes when it’s turned off.