Nice question :)
The main cause of the cancer is the alteration in DNA (which can cause mutations in turn) or any change in the biochemical pathways occurring in a cell/cell system which can alter the lifetime/reproducing capability of the cell.
The stability of DNA helix (backbone) is due to phospho-diester linkages in the molecules and the presence of hydrogen bonds between the A - T and G- C base pairs. The only fundamental force which can interact at the molecular lavel and has the energy sufficient enough to alter the DNA backbone is electromagnetic force. Same applies for the biochemical pathways.
The sound waves are the mechanical waves not electromagnetic. Therefore, the only radiation which can cause cancer is the electromagnetic radiation with high energy, such as Ultraviolet, X-rays or Gamma rays.
Probably not. But, like any scientific statement, it can't be completely disproved.
The use of the term 'radiation' in the question seems to be an attempt to associate medical ultrasound procedures with other tests and therapies which use X-rays, gamma radiation, or radioactive materials. While it is true that we do describe sound as radiating away from a source, the way that sound transfers energy is very different. A noise picked up by your ear might be recognized as a friend who is shouting your name from the other side of the park. This noise travels as a slight motion of the air molecules. These motions of the air push your eardrum around just a bit and, through the workings of the inner ear, send a signal to your brain.
Very loud sounds can damage parts of your ear in ways that will never heal. They do this by breaking parts of cells and pulling tissues apart. Such damage is possible elsewhere in the body where it is likely to cause bruising or in extreme cases even tearing of tissue. Intense shock waves from explosions can kill people even when no material from the explosion hits them.
Damage from ionizing radiation can cause tiny changes inside cells without destroying them. These changes can re-write the DNA sequences and cause the cells to grow in strange ways. Some of these mutations will be cancerous. The cells survive the damage because the change is so small. Sound waves would tend to compress and stretch entire tissues. Individual cells may be torn apart, but not in ways that change just a little bit of their DNA. It's more likely that a cell would be killed entirely or dislodged from where it should be. These physical changes could be painful (like a punch in the arm), but they are not likely to cause abnormal growth.