Both an airplane and a bullet experience the Coriolis effect. The effect isn't caused by the "rotation of the earth" it's caused by the "change in angular rotation of the earth". Both the plane and the bullet start out in the frame of reference at the point on the earth where they separate contact with the earth, wile traveling it moves over a different point on the earth with a slightly different frame of reference. When traveling straight north/south the different angular velocity of the earth causes the plane/bullet to shear right or left, when traveling straight east/west the angular difference doesn't change so it doesn't shear to the side, however, the Eötvös effect then becomes an issue, causing upwards or downwards shear. The Eötvös effect is what was actually demonstrated in the video when the presenter mistakenly called it the Coriolis effect, though they both tend to get lumped together often.
Planes do steer against the Coriolis effect. This is quite small in comparison to the other forces a plane steers against. It doesn't need to be pre-calculated, just compensated for as the plane's flight is often adjusted. No pilot would notice.
Long range projectiles must take into account several factors, such as drop due to gravity, wind shear, air resistance changes because of density caused by different air temperature, thermal expansion due to barrel heating, etc. The Coriolis effect is one of the smaller factors for light projectiles like a bullet. Typical bullet ranges are not enough to matter in comparison to the other factors. However, heavy artillery have much longer ranges and are affected by the Coriolis effect more. This is enough of a factor that the military has tables for artillery:
http://www.dtic.mil/dtic/tr/fulltext/u2/826735.pdfIn particular page 102 provides a compensation table for long ranges at 30 degrees north latitude.
The Coriolis effect cannot be seen in a sink or toilet, they are too small and there are too many other effects that are stronger. Hurricanes, on the other hand, are affected. Opposite spins in opposite hemispheres.
Thork, you can claim it doesn't exist, but you have a lot of evidence and documentation to unseat first. This is a well explored topic and has been demonstrated for many years. This seems similar to when you claimed the power poles across Lake Pontchartrain didn't exist at all or when you claimed Mt. Rainer was next to the ocean. It just makes you look like a nutter when you claim something doesn't exist when it does. The Coriolis exists and must be explained for in the flat earth model.