Specifically, for my Nikon D7000, but also in general. Is this shutter speed basically going to be the same shutter speed as the fastest flash sync speed, or the flash sync speed more affected by things like inaccuracy in when the flash is actually firing / duration of the flash pulse itself?
In essence, the flash sync speed is the highest speed at which both shutter curtains are open at the same time. At higher speeds, the second shutter starts to close before the first curtain is completely open. So for practical purposes, sync speed equals what you are asking for.
There is probably some minimal slop factor involved which allows for a very slight delay in firing the flash (it takes a non-zero amount of time for the flash to pump out all its light at max power, but this should be in the region of 1/1000 of a second or so), but this will not be much. It can be an issue in multi-flash setups though, where you cannot actually use the max sync speed because the slave flashes are just a little bit delayed and won't fire until the second curtain has started to close.
As you suggest the highest speed at which the shutter is at some point fully open will be faster than the quoted flash sync speed, to allow some variation in flash timing, probably just under a stop faster.
I don't have any information for the D7000, but the Canon 1D mkIII has sync speed of 1/250s with most flashes. When a Canon EX flash is used the camera can sync at 1/300s by improving the timing, according to Canon:
1/300s is not the limit however, by further tweaking the timing flash timing using a pocketwizard it's reportedly possible to get the 1D mkIII to sync at 1/500s (with barely noticeable letterboxing), according to Rob Galbraith: