What is the slowest shutter speed I can use for 70-200mm 2.8 with 2x teleconverter?

I'm using nikon 70-200mm 2.8 with 2x teleconverter. Outside on a cloudy day I don't like to shoot under 1/600, actually i like to keep it up around 1/1000 as a minimum just to be safe, since i'm basically shooting a 600mm after my nikon d7000 crop factor, and i'm hand holding it. I use auto ISO and just keep it wide open 5.6 and at 1/600 and i'm happy with my results, the only part i'm not sure about is how slow of a shutter i should use.
When i'm inside or outside at dusk, how slow can i realistically shoot and keep the pictures sharp, with hand holding it? Lets ignore the subject of what i'm shooting, usually just people walking around with no fast action.
Keep in mind, i'm also shooting with the VRII
Any suggestions on how slow i can shoot handholding it would be great help!

Your logic is on the money.

If you use a tripod, all you have to do is capture any movement of the subject.

If you are hand holding the camera/lens combination with the VR activated, then really you are going to have to do some test shots. Why? Because each person has different skill levels when it comes to holding the camera steady when taking a shot.

Start with a shutter speed of 1/1000th second and then reduce it until you get obvious blur

There used to be a rule of thumb like this for slowest shutter speed for handholding…

35mm = 1/35 sec.
125mm = 1/125 sec.
200mm = 1/200 sec.
600mm = 1/600 sec.

See a pattern? Always have a tripod or monopod handy.

And if I do not have a tripod I use "found supports" to rest the camera on or brace my arms/hands against. Such as a post, trash can, wall, doorway frame - anything to steady and brace against.

The distance the light travels in your lens is 200mm… So rule of thumb is 200th of a second… Forget about crop factors and tele factors…

for best results use a tripod or a monopod… Like serious shooters do