Isn't the derivation already available in that link? The group velocity is defined to be the velocity at which the envelope of a wave travels. It's an approximation in the sense that it was derived involving approximating the phase in first order of ##k##.
Yes, the linearization is an approximation which only applies to cases where ##A(k)## is sharply peaked. If this is not the case, higher orders of the phase must be included in the calculation and they usually lead to the modification of the shape of the pulse (so-called "chirping").