Yes, you possibly can use a faster-than 1/250s shutter speed. But not with HSS.
You might be able to use a higher shutter speed than your camera body's maximum sync speed, if you use the YN-622C's "supersync" feature and if your strobe's duration pattern works with supersync. But this is not high-speed sync. This is where the sync timing is delayed until the "flatter" tail of the pulse, for more even illumination as the shutter gap sweeps across the sensor. But it relies heavily on exact timing and strobe pulse duration, and will typically only work with the strobe on full power, and only with certain strobes.
You definitely have to use the PC/3.5mm sync cable to do this, as supersync only works with the YN-622's PC output. And you may have to mess with the YN-622's Fn 01 to adjust the timing. To use it, on the 622 triggers, set HSS.
See also: Clive Bolton's TOYUG (the Other YN-622C User Guide).
On the strobe's specifications it says it supports up to 1/2000s.
No. What it says on the specifications is that the "Flash Duration" is 1/2000s-1/800s. This is how long the flash pulse lasts. It's 1/2000s at the lowest power setting (1/16), and 1/800s at the highest power setting (full power). This is why supersync might work. If you have the flash at full power, the pulse lasts 1/800s. And if your shutter speed is 1/1000 or faster, you may be able to use the tail of the pulse for the entire exposure.