1. Do you call them "tail numbers"?
Yes. Sometimes people refer to them as the "N-number," since aircraft registered in the US have numbers that start with "N."
2. Is the owner of a private aircraft required to file a correct flight plan? If so, would it be noticed that the plane was not following that plan?
If the flight will be conducted under visual flight rules (VFR), then no flight plan is required at all, at least in the US. VFR flight plans serve mainly to alert authorities when a plane is late, and to give them some clue as to where to look for it. If the flight is at/above 18K ft (where a corporate jet spends most of its time), then it will be under instrument flight rules (IFR), a flight plan is required, it will be under ATC control, and any deviations from the flight plan would be noticed immediately. In practice, a corporate jet on a long-range international flight will almost certainly be on an IFR flight plan the entire way so that they can legally fly through clouds, fly at high altitude, and not bump into any other aircraft.
3. If reported an hour after the plane took off, would it be possible to track that plane?
By the timeframe you are talking about (2030), the newer ADS-B (GPS-based) air traffic control system will be fully implemented. Planes equipped with "ADS-B Out" and which have the equipment turned on (mandatory under all IFR flying and some VFR airspace) will be easy to find whenever they are in contact with the air traffic control system.
Also, sophisticated corporate jets frequently use other non-ATC related comm systems to keep in touch with the company HQs via satelite link. These systems may automatically relay info on the acft location, maintenance codes, etc.
4. A commercial flight from DEN to LIM takes twelve hours. How long would it take for a top-quality private jet?
I don't
think they cruise much faster than an airliner. Someone with real experience will chime in. Going faster burns more fuel, and reduces maximum range. The main advantage of a private jet is flexibility in scheduling and some degree of privacy, the time saved in actual flight compared to an airliner probably isn't very much.
Edited to add: I looked up some numbers. The new Gulfstream G500 (a very high end bizjet) is designed for a long-range cruise speed of .85 mach. The cruise speed for a Boeing 777 is .84 mach. So, the flight times won't be appreciably different.
All the above subject to correction by more in-the-know folks.