Another noob question: see the attached image of my thermostat. Would it be possible to switch it for a smart one? What would I have to check? If it is possible, what would you recommend? If is don't want any 3rd party app or account?
That's a Honeywell T87G1006 "On/Off" thermostat. "On/Off" meaning that it can signal to the boiler whether more heat is needed or not. Boilers of that time may also be compatible with the OpenTherm protocol as there's a similar model that also speaks OpenTherm.
The easiest way to go about this is probably to look for a more modern thermostat that's compatible with your boiler but also provides some interface for remote control from your Home Assistant.