Does the SyberJet SJ30 really exist? It is amazing to think about a single-pilot jet with that much range. They have a site, they have some 2 minute videos talking about the aircraft... but there is no video of the thing flying or any review of its interior, no one talking about their experience of owning one... 

Does SyberJet SJ30 really exist or is just a project that never made it off the drawing board?
It certainly seems to exist, at least in the sense that at least one has been built and registered. The Wikipedia article "SyberJet SJ30" contains a picture of one.

The registration number on the aircraft pictured is N7SJ, and the FAA's website says that that aircraft is indeed an SJ30.