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White, Yellow, Green, and Red is all you need for AC and Furnace. 

The furnace circuit board may be acting up. Try a dumb thermostat first to see if issue persists.

Try a cheap dumb thermostat. Your furnace circuit board may be acting up. You can pull the thermostat out and jump the white and red to see if anything happens.

Some furnaces can do so by turning the fan off and on within 1-3 seconds to increase speed. The Carrier can do 5 speeds by toggling fan on and off manually. The 6th toggle will go back to the lowest speed.

You can but then you will lose the communicating part and any other proprietary feature like the Carrier humidity control and etc.