I repeat, do the test below.
TEST: Connect the G and the W wires to the RC/RH wire. The fan should come on and the heat should come on.
1. If the heat does come on the HVAC is fine; if the heat does not come on the thermostat cannot turn it on either.
2. If the heat does come on with the above test, but not with the thermostat connected W to W, R to RC and RH, and G to G, the thermostat is the problem.
3. If the cool comes on when you do the test, that means the W and Y are somehow switched.