A) No, they don't. They often use shotguns modified to be light and pointable if they are professional competitors, but most trap shooters use the same shotguns they would go hunting with. They also use chokes that are modified or less (as in, broad spread, less range) and the patterns are no different than any other commercial off the shelf shotgun (circular). All of my shotguns wear full chokes, which have much greater range, especially in a gun not designed for trap shooting (longer barrel). Also, trap shooters use reduced recoil (AKA, reduced range, reduced velocity, reduced power) loads and their shotguns typically only hold two shells. Basically, this argument boils down to, if a trap shotgun can reach, a hunting shotgun can reach MUCH MORE EASILY.
B) Trap shooters are shooting one shell at a very fast moving target. This guy just had to shoot a stationary target with one of his several shells.
As for your third point, a shotgun pellet at 5 grains and 0.05 BC (typical for a light sphere), loses half of its 1200 fps velocity within 200 yards (600 feet), and does that between a quarter and a half of a second due to aero drag. It doesn't matter which way you shoot the shot, because in that tiny time span, gravity at its very weak 9.8 m/s^2 doesn't affect that hardly at all, as it makes up less than 10% of the velocity change.