Does ground speed represent the horizontal speed measured at sea level? Which of the following vectors is refered to as the Ground Speed?

The green one I expect is TAS, the blue one would be Vertical Rate



Note that for simplicity I did not draw them as geodetic curves, but the black one should be shorter than the gray one because of the earth's curvature. 

Does the Ground Speed represent the horizontal speed relative to surface at the sea level? 

Ex: Helicopter climbing verticaly will have GS=0?

Does the Earth's curvature affect the GS? As the length of the gray vector will be longer the higher the altitude is. Does that mean a plane flying at 1km will have higher GS than a plane flying the same total speed at 10km?

edit:

I'm simulating aircraft movement along a straight trajectory by constant speed. To simplify, the input is start coordinates &amp; altitude and finish coords &amp; alt. 

That means the plane can ascend or descent during simulation. I know the total distance between A &amp; B and distance at sea level. So first thing I tought of was:

GS = distanceAtSL / (totalDistance / speed)


In relation to image:

I know the length of Green line and I can calculate how long it will take the plane to reach the end. The time to finish is constant for each line &amp; I know the length of the black line so using formula speed = distance / time I should be able to get the "Ground Speed".

Correct me if I'm wrong.
(Source)
Ground speed
Ground speed equipment on-board aircraft measure the speed against the ground. Except in the case shown above where distance information is derived from a ground station (DME station).
Curvature
Ground speed is not derived from true airspeed. When using GPS, or an inertial navigation system to measure the ground speed, it will be the speed across the ground. It is the rate of change of the latitude and longitude, so you don't need to include the curvature.
Once an aircraft (or crew) know the ground speed and TAS, the wind information is then known (how fast the mass of air the aircraft is in is moving).
True airspeed
True airspeed is the speed inside a mass of air. In a zoom climb in an F-15, TAS will be pointing almost the same direction as the [upward] nose of the F-15.
If the mass of air is moving due to head- or tail-wind, then this wind speed is subtracted from (or added to) the horizontal TAS to get the ground speed. (But again, this is not how ground speed is calculated, because there is no direct measurement for the horizontal TAS.)
Vertical speed is how fast the plane is climbing/descending. It's measured by sensing the rate of change of the static air pressure.
RE: edit

I'm simulating aircraft movement along a straight trajectory by constant speed.

This is not how airplanes fly in real life. But for the sake of your model, and if you don't have wind simulation, then the known green distance and the time it takes to move from A to B will do just fine to compute the ground speed.