That's kind of picky to ask a thermostat to be that accurate. Two degrees is awful close to the set temperature.
You may be thinking of the old analog thermostats that had the bimetal coil. Modern digital thermostats should be accurate to within one degree. By accurate, I mean they should turn off the heater when the temperature exactly matches the displayed setting within a degree or so. There are upper and lower thresholds that prevent the system from constantly cycling on and off for fractions of a degree.
Our thermostat stops heating or cooling when the temperature on the display matches the setting. It fires up when the display shows a degree (rarely two degrees) off from the setting.
Most likely, @Quickstep
's thermostat needs to be replaced. I have replaced failed thermostats in at least three homes. I bought digital ones at the big home store. They were middle grade models - not too fancy or complex, but with sufficient features to meet our needs.
Fans on modern thermostats usually run slightly longer than the actual heater or air conditioner, to purge the warm or cold air, so it doesn't sit and go to waste in the ducts in the attic (or wherever the ducts are). This is an automatic function of today's thermostats.
Most new thermostats run off the power in the thermostat wiring - no external power needed. In a typical installation, you connect four wires, configure the thermostat for the type of heating and cooling in your home, set your times and temperatures, and you're done.
I just checked our thermostat, and we have a Honeywell RTH7600 model. I don't remember when I had to replace it, but my spouse says that it has been well over five years ago. According to the installation manual, it controls a long list of system types, too much to type here. That includes heat pumps. I don't know about the variable fan speed, but you can ask Honeywell at 800-468-1502 (from the front of the installation guide for wiring help).
Unless you have something very unusual, modern thermostats should work with most home heating systems. They are easy to install and relatively easy to operate. They are designed for ordinary homeowner installation. They practically beg you to call the company first and not return the thermostat to the store if you run into problems.
Placement of the thermostat can make a huge difference. My parents moved their thermostat from the living room to the hallway. The living room location was too close to a lamp, back in the incandescent bulb days. Heat from the lamp affected the thermostat. It was a pain to adjust the thermostat when you turned the lamp on and off, so I helped them move the thermostat.
Thermostats are getting more sophisticated, and arguably easier to use, as manufacturers deploy new technologies that sense when you are home, what temperatures you like and when you like them, etc. In addition, they use more sophisticated programming to improve overall comfort and attempt to save power.
This is a personal statement:
I do not recommend internet-connected thermostats, such as the NEST. For me, this is a privacy and security issue. Others do not seem to care, or do not understand the issues, which are far off-topic for a woodworking forum.