Is it true that calcium supplements contribute to clogged arteries in a person that already has heart disease?

Poor evidence. Not really. There was one poorly done study and the evidence that calcium supplements clog arteries is pretty weak at best. It hasn't been show by any other studies. In any event, take only the calcium you need and get most of it from your diet if possible.
Possibly. Read my blog for a good overview of the subject.
Unusual. There is no true cause effect relationship. Taking calcium supplement daily at regular doses without causing hypercalcemia should not cause increase risk of coronary artery disease . In special situations when patient has chronic hypercalcemia for years some reports found more calcium deposits in heart valves and calcium in the coronary arteries wall causing some narrowing , but only in chronic.