Answer 1:
Dont get confused with the difference between airspeed and ground speed. For the Pitot tubes which I described to you earlier, the speed given is the airspeed, or the speed relative to the surrounding air. The ground speed is the airspeed corrected for wind effects. If you were traveling at a constant airspeed, say 500 mph (which is what you'd read from a Pitot tube measurement), then when you're going into a 20 mph wind, your true ground speed would be 420 mph. If you're instead going with the wind, you'd be at 520 mph. Let's look at this in more detail.
In physics, objects in motion must be examined with relative to other objects, or their surrounding medium. In this case, examining the Pitot tube determines the speed of the plane relative to the surrounding air. So let's look at the case of a plane going into the wind. If the plane was standing still (airspeed = 0) but was facing the same 20 mph wind, you should see the reading as 20 mph. It doesn't matter whether the plane is moving 20 mph in still air, or if the plane is not moving with a 20 mph head wind  it's treated the same way mathematically! Now, if we increase the speed of the plane until we read an airspeed of 500 mph, it is clear that our ground speed would only be 480 mph, because 20 mph of that was from the wind.
From a more technical standpoint, what we are doing here is defining the "frame of reference" of the problem. We can imagine the frame of reference to be moving with the wind. For the case above, the frame of reference would be moving against the plane at 20 mph. Once we determine the airspeed is 500 mph, then we'd subtract off the frame of reference, which is 20 mph to get 480 mph. For the case of a tailwind, the frame of reference would be moving with the plane at 20 mph, and by now adding that, we'd get 520 mph.
The limitation here, of course, is that by using this simple measurement, we cannot accurately determine the ground speed. The only way to do that would be if you knew what the wind speed was at all times outside the plane. To know the true ground speed, you would need fancier measuring devices such as GPS (as you mentioned earlier) or, I think they can also use some type of radar (Doppler effect) to get true speed as well.

Answer 2:
On an airplane, there are two speeds (possibly in different directions) that are measured. The first is airspeed which is the velocity of the plane with respect to the air it is flying in. The second speed is ground speed which is the speed of the airplane relative to the ground. These two velocities can differ a lot  and may not be the same direction.... For example, a plane with an airspeed of 200 nm/h into a direct headwind of 50nm/h will have a ground speed of only 150nm/h. On the other hand if the wind is from the side of the aircraft, it would have a ground speed of 206.2nm/h, but in a direction that is 14 degrees away from the plane's centerline. i.e. it is drifting sidways, but that is not apparent to the local windflow around the plane.
To detect the real ground speed, there are several ways, the most popular is GPS units which have revolutionalized navigation as they always display the true ground speed and heading, independent of the air motion. Other systems are the Loran and Vortac systems which supply direction and approach velocity information to sites on the ground. Lastly there is seat of the pants flying where you estimate the wind aloft speed from weather reports and use the local airspeed to navigate. (This is also why some allied bombers in WW2 droped their bombs in the channel, as it is highly inaccurate.
So  the airspeed measures the speed of the plane relative to the air, while the ground speed is relative to the ground. In a jet stream, the ground speed will be heavily modified, while the airspeed will change only if the aircraft adjusts drag or throttle.
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