And for the energy conservation, why will they have same velocity? Shouldn't it be different because they have different amplitudes? And my question exactly is why energy flux is conserved and not energy?

And for the whip, the right side of the boundary is taken to have zero impedance which implies free end and also that the linear mass density is zero to the right. But the left side of the boundary is still present and when you do the math, it tells us that the amplitude of the transmitted wave is twice that of the incident wave. But my question is, the physics that we have built these concepts on till now tells us that no particles in the medium implies wave cannot travel in the medium (because wave consists of infinitely many simple harmonic oscillators). How can one explain this?

I understood your point.

If the y-coordinates are the same then, shouldn't it be $A_1$=$A_2$ only.