That is my question.
Well . . . . . . yes.
For one thing, if nothing else, the pilot is generally watching something on the ground while hovering. Why else would he stop? He just needs to take a break or something? The pilot would automatically move the craft to adjust for wind, etc.
‘Hovering’ is defined as stationary relative to the earth, so quite simply yes.
What’s actually happening is they’re stationary relative to the athmosphere, which in turn rotates with the earth.
Except if there’s a breeze. In that case, the air is moving past the helicopter while the helicopter, by definition (as you pointed out), remains stationary relative to the Earth.
Definition of hovering is appreciated. Let me rephrase: a helicopter ascends 10 feet in the sky and then the operator takes his hands off the controls. Will the helicopter stay in the same position relative to the earth? Or will the earth move “out from under it”?
Well, yes, I was simplifying. (But what if we define breeze as air movement relative to the athmosphere? 
)
Speaking of relative movement, on Star Trek Picard will say, “All stop!” So they stop the ship… but relative to what? Without a reference point to be stationary from, speed is meaningless.
The helicopter will move, but not for the reason you’re thinking. Helicopters are inherently unstable. If you let go of the controls, you will crash.
To be more serious (not that letting go of the controls and crashing isn’t serious), the Earth will not “move out from under you”.
From what I understand of 'copter flying, once that happens, the heli is going to crash.
Gawl dang it!! 
Assume there is a helicopter that, once it ascends, it will not crash if the operator takes his hands off the controls. There is a big red spot directly under the helicopter at this point. In five minutes, will the red spot still be under the helicopter??
Damn, I wish I’d said that!
[sub]Oh, the comedy![/sub]
Fiddle Peghead: If you’re riding a train and jump straight up, does the train move from under you?
The Universe?
Sure it’s all moving, but you’d think they’d be able to work out an average of ‘stopped’ by then.
Besides. I think it is possible for there to be an absolute where something can be not moving where relativity is irrelevant. (Imagine emptying the universe of all matter, and then placing one object in it. I think it is possible for that object to be moving or not moving. But without other objects it would be impossible to find out)
That would require autopilot technology. Therefore, it’ll do whatever the autopilot is designed to do. The problem is that a helicopter is dynamic (constantly self-adjusting its position) - and it does not provide a fixed point from which to measure anything.
I presume this is the kind of problem you’re wanting to talk about - a helicopter introduces too many variables to get a simple answer.
Well, no. That is the essence of my question though. Is the helicopter affected in the same way that a thrown ball in a moving car is affected?
Exactly. And that defines the concept of “relative motion”. If motion weren’t relative, you’d be able to say with certainty whether your hypothetical object was moving.
Yes. A helicopter will remain above the point just as will a person jumping straight up on a train or a ball thrown straight up in a car, and for the same reasons.
Well, yes and no. If there is zero wind relative to the ground below, you’re correct. However, unless corrections are made to account for wind, the 'copter will tend to drift in the direction the wind is blowing.
That’s true. I was assuming a no-wind situation, since we were “tweaking” the definition of hovering. If there is wind then we have to go back to the definition of hover, which requires control inputs to keep the aircraft over the spot and to keep it from crashing. But in a theoretical situation (“Does the Earth move from beneath a helicopter that rises straight up from it?”) there is no wind.
Helicopters will also “transition” due to the thrust froj the anti-torque rotor. That is, the engine produces torque. In U.S. helicopters the rotor turns anti-clockwise (left); thus, the opposite force makes the airframe rotate clockwise (right). You need to add left pedal to conteract the torque. But this means that there is thrust to the right, so the helicopter will tend to drift right. This is corrected to some degree by A) rigging the rotors so that they tilt a little to the left, or B) tilting the rotor mast to the left. But you’ll often still get drift that requires pilot input.
Good points. So, the answer is “yes, but”?