How can planes land on a rotating Earth? If the earth is rotating (e.g. at 1000km per hour, at the equator), how can planes safely land on a moving runway?
 A: It is the relative velocity between the plane and the Earth which is important.
When the plane is at rest on the runway it is moving at 0 km/hr relative to the ground but also it is moving at 1000 km/hr due to the rotation of the Earth.
So if the plane is coming it to land at 150 km/hr that is 150 km/hr relative to the ground.
If a plane needs to travel due North then it does have to compensate for the rotation of the Earth and so must fly on a heading which is West of due North to arrive at a location where the speed of the Earth's rotation is less than 1000 km/hr.
A: The air, and the planes in the air, are rotating together with the Earth.
Everything is rotating!
A: The fact that the Earth is rotating means that it is rotating around its own axis relatively to the surrounding space. The Earth's atmosphere – even though not opaque like the rocky planet's surface itself – is just the outmost layer of the planet. It is not a nonchalant part of the outer space. 
When the Earth orbits the Sun, it does not shed the atmosphere into the empty space all along the orbit, because the atmospheric layer belongs to the Earth. 
Take Jupiter, a gas planet: There wouldn't really be any tangible surface to land onto. Basically all you can see from outside is gas, yet all of that moves along with the planet, as one unit.
A hypothetical but impossible example: If only the Earth rotated at 1667 km/h but the surrounding atmosphere*  didn't, the velocity is a relative one: While standing on the Earth's surface, you should basically be standing amidst a raging storm wind of 1667km/h, which isn't happening. Do not mix this hypothetical example with the weather system, which is an entirely different concept happening for other reasons.
*(which isn't a void vacuum, but a composition of gasses, therefore, matter which can interact with other matter. Colliding into gas molecules slowly doesn't feel like anything, but the bigger the velocity of collision, the more dramatic violence you can have mundane seeming particles cause: Still air can turn into house-tearing hurricanes when moving fast enough relatively to the houses.) 
If you're traveling in a train and you jump, you land back where you were just like you hadn't even been in a moving train, because the air inside the train is traveling with it.
If the train has no walls, it isn't carrying a portion of still air, but is slashing relatively very fast through the surrounding air. If you jump up now, the passing air will hit you hard and you'll fly far backwards relative to the train's floor.
The Earth with its atmosphere is like the closed train that has walls.
Landing onto a rotating Earth seriously is a concern when astronauts return from space, because space is an airless zone and when you enter the Earth's atmosphere, the rotation of the planet suddenly starts to affect you. Similarly, when you launch a rocket to space, you can't just point it wherever you like, but you have to consider the Earth's rotating and orbiting factors. A bit like being in a moving train, opening the window and trying to throw a ball into a bucket that's sitting still next to the railway track. 

But the train travels with a linear motion. So, as regarding the earth's rotational motion - if the train was shaped like a sphere (for example), and its motion was to rotate around its own centre (axis), would passengers not sense some sort of centrifugal force? – tmccar

Newton's laws of motion state that "When viewed in an inertial reference frame, an object either remains at rest of continues to move at a constant velocity, unless acted upon by a force". When you're in a fast carousel, you feel like you're being pushed outwards from its center. This feeling of pressure happens because according to classical mechanics, once set to motion into one direction, you'd really want to continue going that way in a straight line, but the supporting physical obstacles – the carousel's structure itself – blocks you from following that natural path, reflecting you back towards the carousel's center. You're getting a fight of two forces. 
When you roll a ball at a straight wall in a diagonal angle, it will reflect back away in the identical but mirrored angle. You can observe that one reflection. Inside a circle, you'd want to just go straight out but your trajectory is being reflected back inwards at an extremely subtle angle at a time, and because the obstacle is curved, the moving object barely gets to travel on its newly reflected path at all when it already hits the circle wall again and is reflected time and time again, resulting to travel in what appears to be a circular path. 
If you had a rotating sphere full of air, the air molecules near the edges would be more likely to occasionally collide with the wall, so there could be some turbulence there, but the air in the middle of the sphere would remain more untouched. If you were to have two spheres nested so that there's an outer shell sphere, and an inner smaller sphere which contains air – if the outer sphere rotates around its axis but the inner one that is in contact with the air inside doesn't, the inside remains fully intact and ambivalent of what's happening outside. If the space between the two sphere shells is a vacuum, there is no matter that could transfer the outer sphere's kinetic energy to the inner sphere. 
When a normal shaped train with air inside is in motion, the inside walls themselves are not moving relative to the air inside, so no air bouncing occurs. However, if you're standing on a train platform where the air is standing still and suddenly a train passes by, its walls hit the air molecules, causing a blast of wind.
