As the ice melts, there isn't a change in water level. For every gram of ice that melts, you're displacing one less gram of water, but another gram of water is added from the melting ice. In other words, no change.
Incorrect. While mass of the systems remains the same, the volume changes. 1 gram of water takes up less space than 1 gram of ice(which is why it floats).
1.If not for the stone, the water level would remain constant as the cube melts, due to the excess volume of ice being above water, and so, not contributing to the water displacement.
For the sake of easiness of calculations, let's assume that Vi(volume of ice cube)=2liters, Di(density of ice)=50%Dw(density of water). It is unimportant for the resolution of the problem that the real water vs ice density ratio is different.
2 liters of ice weight as much as 1 liter of water. However, only 1 liter of ice is submerged, with the rest sticking out above the water level. As it melts by half, it adds 1/2 liter of water, while shrinking by 1 liter, of which only 1/2 was under water. The amount of water added and the amount of volume freed is the same, so the water level remains constant.
2.Now, if there is a stone in the ice cube.
Let's assume it weights just as much as 1 liter of water(1kg). In this case, the cube is fully submerged. As it melts by half, it adds 1/2 liter of water, while shrinking by 1 liter, of which 100% was under water. The amount of water added is 1/2 of the vloume of space freed by melting, so the water level goes down as the cube melts.
3.When the stone is released, the remaining ice resurfaces, putting half of its volume above water, causing the water level to fall again.
As it settles on the surface, we've got the same situation as described in 1., meaning that the water level remains the same.
The answer is: Water level will fall, then remain the same.
If you want to play with real densities of water and ice, you'll get the same answer.