Is it safe to run with peripheral artery disease?

Yes. Yes. As long as you can tolerate the pain, any form of regular exercise is actually a good thing and will increase your endurance.
Maybe. It depends on how severe your pad is. If you can only walk 4 or 5 blocks before your legs start to hurt, then it is probably a real bad idea to run. Pad causes decreased blood flow to the legs. If you run, you need more blood flow. If your disease is only minor, you may be able to run, but I would ask the physician that told you you had pad.

Related Questions

Is it possible that my peripheral artery disease is just neuropathy?

Symptoms different. Pad is atherosclerotic narrowing of arteries. As it worsens, blood flow decreases leading to symptoms of claudication (pain - walking or activity) which is relieved by stopping. Symptoms change to pain at night awaking you from sleep & relieved by getting your feet down. Finally they progress to constant pain at rest with or without non healing sores/ulcers. Diagnosed accurately by doppler.

Is peripheral artery disease correlated with pain and heaviness in your legs?

Potentially yes. Peripheral vascular disease causes three levels of symptoms as the disease progresses, 1) claudication; muscle cramping with exercise which also resolves with rest. This is very reproducible 2) rest pain; pain when elevating legs / lying down, 3) tissue loss; foot ulcers / gangrene / non-healing wounds. Pain and numbness are more frequent than heaviness.
Yes. Pad can cause this but venous insufficiency can also. Better get checked.
Don't forget veins. As dr. Aalami said, heaviness is unusual and in my experience is more likely to be associated with venous insufficiency. I agree with dr. Korna, it is worth checking it out to make sure that veins aren't the cause of the problem.