The air is blowing inside the house - but warm air... at the unit outside I can hear a humming noise, but the fan is not on. What is most likely wrong with the unit, and about how much should I reasonably pay to get it fixed?

It sounds like the run capacitor or the motor itself in the outside unit is defective from your description. I am sorry but trying to give you a price is very difficult. It all depends upon the parts markup and the labor rate in your area which I have no Idea about.

I would also like to add that the humming noise could also be a item called a contactor. If that is the case a blown fuse or tripped circuit breaker for the outside unit could be a cause. The contactor is closed causing a slight hum but no power is passing through to allow the unit to run.