OpenTherm Thermostat

Stock firmware is not open source, but you can write your own firmware adding new cool features that fits your needs the best. DIYLESS Thermostat is based on Espressif's ESP32 Soc, which enables BLE/WiFi connectivity and allows writing a firmware using Arduino IDE, VS Code or IDF (Eclipse) IDE.
Also you can use ready-made firmwares like ESPHome, Tasmota etc.

If the stock firmware fits almost all your needs and only a few feature(s) are missing - we can consider adding them to the next version, please report a request or an issue report into our thermostat repo. Subsequent updates are free, and if you experiment on your thermostat with a custom firmware and later decide to rollback to the original one - we have a simple browser-based tool which allows you reverting back to the stock easily.

We use a PI algorithm which calculates the needed boiler power level based on a difference between current and target room temperature and also takes into account the speed at which this temperature rises/drops. It is aimed for the best comfort, meaning less temperature fluctuations.
That means the boiler will spend more in active state however at a lower power level instead of cycling between max power and off.

No, with a current control algorithm implementation you would not need a knowledge on outside temperature, because thermostat senses an internal temperature and once it drops - boiler will rise radiators temp to compensate.
In practice (from a real installation) - room temperature accuracy is held within a ±0.15°C range.

Yes, it does. However to use it properly - you will need to distance it from a board using a wires, since ESP32 self-heats a lot and sensor is heated by the board.

Out of the box it is controllable within your home network. If you need world-wide controlling options - one of the possible ways to achieve this is a Home Assistant with its companion app.