How much does Earth's rotation affect flight times in going East or West, and how could you calculate if zero winds?
It does not affect as much as one would think.
The atmosphere moves along with Earth's rotation. An airplane taking off from one place on Earth has to move through the atmosphere which is moving with Earth.
Now you may ask when does it take longer when we travel from Western Europe to Eastern USA, compared to USA to Europe? That depends on Jet Stream:
Jet streams are fast flowing, narrow air currents found in the atmosphere of ... Earth.
Many air routes take advantage of jet streams, as you can see as follows:
Pilots know about the jet streams and will try to gain benefit of them. Recently, there were some news that a flight from New York to London traveled faster than expected because it was in jet stream (details here and here).
If you want to know why Earth's atmosphere moves with it, please see this.
There is a similar question on Physics.SE.
It affects flight time the same way that it affects driving time.
Cars measure their speed over the ground based on wheel speed. Airplanes can measure their speed over the ground using things like GPS. Either way, it's referenced to the ground. Technically the ground is moving, but it's not perceptible to us and we still would say zero speed means not moving over the ground.
Airplanes primarily measure their speed through the air (airspeed), and the air tends to move with the ground, so the situation is the same. If the air is not moving the same as the ground, it's called wind, which is the primary influence on flight time. Wind is influenced by the earth's rotation, but the rotation doesn't directly influence flight time.
This is a physics question, not an aviation question, and might better be posed on Physics StackExchange.
Simply put, the question has an infinite number of answers! Let me explain.
When you are stationary in a helicopter, waiting to take off, with reference to an observer also standing next to the helicopter, your speed is 0.
But with reference to an observer in a static point in space whose frame of reference is the centre of the earth, you are already moving at about 1040MPH. The surface of the earth is also moving at 1040MPH. (BTW, it is itself impossible to define a static point in space since all of space is moving.)
The atmosphere you are about to fly in (assuming zero wind) is also moving at the same speed. If its speed was zero, with reference to the stationary observer in space, then the 1040MPH wind might make flying a little difficult.
You lift into the hover. Now to both observers, you are still doing the same speed because speed is a scalar and you have added no force to change your horizontal speed. To the observer standing near the helicopter, the earth, the helicopter and the air it is flying in are all moving at 0 speed. To the observer in space, they are all moving at 1040MPH.
You now fly West at 100MPH. To the observer standing on the earth, your speed is now 100MPH but to the observer in space, your speed is now 940MPH. You have slowed down!
By picking any arbitrary observer, at any point in space, and doing any speed you choose, you can see that there are an infinite number of answers to the question.
Only by asking the question with reference to a specific observer, can you answer the question.
If you ignore wind and side effects such as the Coriolis effect, the earths rotation has no effect on your journey time since you and the craft you are in must move at the same speed within the same frame of reference, relative to any given observer, together.