I know nicotine is a vasoconstrictor--does raise the heart rate?

Yes and no: As nicotine constricts the coronaries, it also increases vascular resistance to blood flow. For the heart muscle to receive adequate blood flow , it has to increase heart rate so that per unit of time and rate, diminished blood flow is compensated. Subsequently, cardiac output increases and blood pressure will also increase. A severely high BP may in time slow down heart rate as a reflex mechanism.