It doesn't sound like your heat pump is working correctly to me.
If you refer to the attached diagram, the basic heating mode operation is such:
- Outside air is used to warm the refrigerant
- The refrigerant releases its heat inside the house
- The cooled refrigerant is returned outside to be rewarmed
The surprising thing is that you can actually get heat from the outside air when it is 40 degrees, but it does work. However, when the outside temperature drops too low, there is not enough heat to be had, so there is also an electric heating element in the inside unit to provide heat in this case. On your thermostat, there should be some lights labeled "Emergency Heat" or something similar. If these are on, it means the electric element is being used--which will boost your electric bill.
You also said the blower does not run. This sounds like a big problem (and may be the root problem). While the inside and outside blowers will act somewhat independently, they basically run at the same time.
Another problem you've noticed is that the outside unit runs regardless of whether your thermostat is on or not. This sure doesn't seem right.
Since this is an apartment, I'd definitely notify management and get it fixed. In addition to mentioning the sky-high electric bills, I'd tell them everything you've mentioned here and say that you think it could be damaging their equipment--costing them more money in the long run.
Best of luck,