I've also noticed that the thermostat can read 18 degrees in the summer, and we feel nice and warm. Yet in the winter, the thermostat can read 20 degrees and we feel chilly. So, something more than temperature is at play. I am guessing at humidity.

It might or might not have to do anything with humidity. The temperature sensor is measuring the temperature in one location, not the whole living area. Let's say for the moment it is mounted on an internal wall. That would mean, in summer the temperature near an external wall far from the sensor is possibly higher than at the measured location. And in winter, the opposite is true.

(this paragraph is mostly a general explanation for future readers, OP seems to be already considering how this): Modern heating systems sometimes come with multiple sensors to get a better average. Also, i have seen more and more heating systems that also install one or more outdoor sensors (especially central heating systems that circulate hot water through radiators). The external sensor allows those systems to compensate for the time it takes them to warm up the water by detecting the dropping temperatures before the chill has time to seep through the walls.

A temperature controller (just as most other regulation systems) also has to have a deadband ("switch off when reaching 20°C, switch back on when falling below 19°C") to reduce the on/off cycles and prevent prematurely wearing out the system. This deadband can also account for perceived and real temperature differences. For more complex control loops, you might have to read into PID controllers and such.

As for your project, if you only need to read temperature and make it available over the network, an ESP32 controller (Microcontroller with Wifi) with a reliable temperature sensor like a DS18 could be a way to go, especially if you need multiple of those.

Especially for stuff running outside, i strongly recommend not to use mains voltage directly. For these kinds of projects i usually run 12V with a proper fuse (car fuse holder+fuses are fine) at the source! Run the 12V outside, then use a step-down/DC-DC converter next to your electronics to step down to the proper voltage. This way you won't get electrocuted if something goes wrong, and you won't have to worry about drooping voltages in the long cable runs.

If you like to design your own PCB to host the DC-Converter, the pinheaders for the ESP32/Arduino/whatever and connector to the temperature sensor and power lines, you can freely copy stuff from my NDC1 modular Arduino based computer project. The (KiCAD) schematic for the DC regulators on the PowerBoard is pretty much my goto design for most of my electronics projects and has been tested extensibly under outdoor conditions in the range of about -25°C to +60°C (no guarantees though, it's a hobby project after all).