1 Why are the handheld photos (with VR on) below 1/60s consistently sharp and between 1/80s and 1/125s showing double contours?
The vibration from shutter and mirror has the largest amplitude right after the shutter opens. Over time the vibration amplitude decays. At longer exposures, the largest amplitude vibration affects on the beginning of the exposure so most of the light that is recorded renders the image sharply.
Mirror vibration and shutter vibration probably have different frequency ranges.
So, if you use a solid tripod, EFCS and M-UP you should get sharp pics at all speeds in most conditions (wind may still cause vibration). If you need to hand hold, you should preferentially use a fast shutter speed which takes care of subject movement as well as all kinds of camera shake.
2 Using flash @ 1/100s with VR On handheld does not show double contours. Why is that?
The flash duration is about 1/1000s or it can be even shorter. Thus the vibration doesn't have time to make an impact on the image sharpness.
3 It seems to me that the issue of double contours is caused by mirror shock. Is that right? and if so why don't I see that on my Nikon 70-200 f/4?
The 300mm has greater magnification due to its focal length so any vibration is magnified as well. The PF is also a very lightweight lens for its focal length which can accentuate the problem. Less mass, easier to vibrate.
I typically use the 300 PF for photographing moving subjects, mainly people, at concerts, fashion shows, and sports. This is hand-held work. I also occasionally use it for landscape and close-ups (most of the time on tripod, but occasionally hand held with VR).
My most typical shutter speeds for the moving subjects are from 1/500s to 1/1250s, with VR set to SPORT mode. This means the lens won't fight sudden changes in subject trajectory as it would in NORMAL mode, and it is easier to keep the AF sensor squarely on the subject's face during the action than with VR OFF, and the viewfinder is more stable so I can get more precise compositions. I do not really notice a decay in sharpness at these speeds due to having VR in SPORT mode. The benefits outweight any drawbacks in my case. If the subject is moving slowly or I am shooting indoors in low light, I may drop the shutter speed to 1/250s or so; this has worked out well but sharpness is typically better at faster speeds. If I have to shoot at 1/160s or slower, I don't have much in terms of expectations. I fire a number of frames in that case and hope to get an acceptable one. This is just as much to subject movement as it is to imperfect stabilization of the lens. A heavier lens such as my VR 200/2 II gives some additional leverage in shooting in low light but even with that, slow shutter speeds often yield images which are affected by subject movement if I go to speeds of 1/100s or so.
I rarely use VR NORMAL on the 300/4 PF. It is intended for relatively static subjects and I prefer to use a tripod for most shots of such subjects, to get guaranteed sharpness. For tripod use I keep VR OFF.