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?
This suggestion depends on your skill level and desire to be completely free of corporate bs.
The ‘smart’ is already in home assistant. You do not need a smart thermostat at all. Instead you need temperature/humidity sensors and actuators for your HVAC system. There are several decent smart climate control integrations already and a library for PID control if you want to make your own control automations.
Of all the smart things out there, a thermostat is completely unnecessary. The only solid reason to buy one is convenience, if one is just not into thinking about it too much.
If privacy and autonomy are a priority, however, this is low-hanging fruit to declare independence from corporates who like to rummage through your underwear drawer.