Exactly -- they don't operate as taxis; they operate as "limo" or black car services. The rates are negotiated, often with direct input from the driver.
That depends on state law. For instance, in New York there are yellow taxis that you can hail but you can't call (on the phone) to be picked up; green taxis that you can call on the phone to be picked up in most of the city; and black cars that you can call on the phone for a pickup anywhere in the city, but which don't have to follow the same rates as the yellow and green taxis.
In Reno, you hardly ever see a taxi driving, so they can all be called for a pickup. I honestly don't know if one would stop if you hailed it.
Aside from real limos, I can't remember seeing luxurious taxis outside of New York. But if they exist, it would be because the local laws allow them to charge a different rate than regular taxis.
Minor correction: The green cabs ("boro taxis") can be either called or hailed, but in either case can only make pickups in the outer boroughs (and the very northern end of Manhattan).
