OK, I'm sure you now know what thre rewind collar is and does. By "frame index" I mean how far the film is allowed to be wound when you wind it on. If the egars are stripped or some othe r damage is done, you may find the advance is erratic.
As I said the first thing to do is ascertain double exposures of what?
The best way to check the film transport would be to run a roll through another camera. Then you develop it but keep it in a long length. With luck you already have a film available. You then roll it back in a cassette and load it into the camera in the normal manner and fire a few frames with the back off. You can then rewind it just so much that a negative is properly aligned in the frame aperture. You then have a proper reference as you wind and shoot, so you can check each frame is over the opening in the kosher manner.