1/1000 s for a strobe duration is really long.....10 times shorter than that is more likely. Any blur is almost certainly due to ambient light and camera settings that allow that ambient light to get to the sensor. Use small apertures, low ISO, strobes on high power, and avoid any camera automatic exposure settings like P or A or S.
I don't think so.
To get to near 1/10000 of a second one has to turn down the power. For example-
This site has some actual measurements:
http://gock.net/2012/01/flash-durations-small-strobes/
The first strobe listed is a Canon 580EX
At full power the duration is just 1/250 s.
At half power it is close to 1/1000 s.
To get to near 1/10000 s. one has to go to 1/32 power.
Other strobes are not as quick.
Both Broncolors had longer than 1/1000 s. duration at all settings tested.
It would be cool if we had data like this for underwater strobes!
I typically use my underwater strobes at 1/2 to 1/4 power so I expect the flash to be closer to 1/1000 s. rather than 1/10000 s.