Why do airplanes experience negligible Coriolis force while bullets experience the Coriolis force in long range shooting, even though the mass of airplane is much bigger than a bullet?
Just to make things simple, suppose you are standing at the north pole, and you shoot a bullet south at some speed, aiming for a target 1 km away. In the time it takes the bullet to get there, the target has moved east a certain distance, because the target travels in a complete circle around the north pole in 24 hours. From the viewpoint of the shooter, who is turning with the earth, it looks like the target is standing still but the bullet has curved, but that's just a perception.
Now suppose you ride a train, on tracks fixed to the earth, from the north pole to the target. In the process, you will pick up the eastward speed, and you will experience that as an acceleration toward the east. That's coriolis.
Now suppose you are in an airplane, making the same journey. Let's assume there is no wind, that the air is fixed to the earth the same way the train tracks were. Well, the plane travels through the air, so as the plane travels south, the air will carry it to the east, giving it that eastward velocity. Coriolis, again.
The acceleration felt is just proportional to the southward speed of the train or airplane.
Why do airplanes experience negligible Coriolis force while bullets experience the Coriolis force in long range shooting?
You are confusing the force with the consequence of the force. Consider a powered parafoil whose total mass is a mere 100 kg (motor+parafoil+pilot) and is moving at a mere 25 km/h and a 50 caliber bullet whose mass is 50 grams and is moving at a hefty 3125 km/h, both at the same latitude and both heading in the same direction. The Coriolis force on the powered parafoil is 16 times that on the 50 caliber bullet.
On the other hand, the consequence of this force, the Coriolis acceleration, is much greater on the bullet than on the powered parafoil. Now mass doesn't come into play; it's just the ratio of the two object's speeds, which is a factor of 125 in favor of the bullet.
Airplanes always maneuver with respect to the surrounding air. Something which confuses beginner pilots is the following question: imagine you have a wind from north to south, and you point your airplane to the west. Where is the air pressure higher?
a. The left side of the plane b. The right side of the plane
The answer is: neither. The pressure is the same on both sides, because the motion of the airplane is best described in a frame where the air is stationary. The pilot can only detect there is a wind by noting that the trajectory of the plane is veering south.
So the answer is: yes, Coriolis forces do affect airplanes, at least as far as winds are affected by Coriolis forces. Whether or not Coriolis forces on the plane are important depends on how long the journey is. See here, for example. It is true however that a pilot doesn't have to know anything about the Coriolis effect in order to get to their destination, assuming they're not particularly short on fuel: a plane is subject to course deviations all the time, most of which are much more important than the Coriolis effect, and a pilot can simply correct for them by pointing the nose in the right direction.
Planes and bullets are both affected by Coriolis effects. The bullet will be more affected than the plane, because it's velocity is higher, but that's only the tiniest part of the story.
Bullets are unguided projectiles. They go where the physics says they go. Any effects due to Coriolis are unabated. Also, bullets which are fired at distances where Coriolis really start to matter are also trying to hit a very small target. Small effects are very noticeable.
Planes are controlled. There is a guidance system (such as a pilot) which is trying to keep it on course. This is important because there are forces that will deflect the plane off course which are orders of magnitude more powerful than the Coreolis effect. If there was not active correction, those other forces would virtually guarantee that we never arrived at our destination.
Because there's a guidance system in play, we rapidly trim the aircraft to counteract the Coreolis effect as a side effect of trying to trim the aircraft for all of the much bigger forces at play. There may be a slight deflection of the rudder associated with the Coreolis effect, but you would be hard pressed to identify it amidst all the other forces.
A lot of what people don't understand is that the airplane is going to be affected more by gravity than a bullet. The airplane doesn't have to correct the nose of the plane because of the physics of the craft along with the earths gravity pulling it downward. So the plane is basically orbiting the earth and that is why the earth doesn't fall away from the tip of the plane with coriolis effect when plane is heading east. The planes physics along with gravity keep the entire plane level essentially, keeping the craft airborne and also keeping the nose at the same level as the tail of the craft in prospective of the pilot. So if the plane is heading east, the ground will not fall away from the plane the tip of the plane is pulled downward already because of physics. A bullet however isn't affected as much by the same force the plane is experiencing, resulting in a elevated hit on the target (if facing east) and a lowered hit on the target (if facing west) because of the raising of the target or the drop of the target and not the drop/raise of the bullet.