My outside unit fan (on top) runs constantly. The only way it shuts off is when turned off from the main breaker box. Plus, I think the unit is running unusally noisy and sometimes a very high pitched whinning is noticeable even from inside the house.

Can I run the heat without the air compressor being turned on? Will it hurt anything?

Is this a heat pump? If so, do you have backup gas or electric heat? If you have backup heat, yes, you can leave the outdoor unit off and just use the backup. If no backup, then you will get no heat without the outdoor unit running (if heatpump system).