Ok, although the documentation is not necessary for elementary steps, you know.
No, we don't. I don't see where you get this out of your source. Could you indicate an exact quote ?
What we have is that convection gives rise to a certain adiabatic (equilibrium) lapse rate. But nowhere we see anything about a relationship between the RATE of convection, and the lapse rate (= change of temperature with altitude) itself.
So I bug already here....
What is true however, is that the further the actual state of the atmosphere is from its stable state (with the adiabatic lapse rate), the more instable the atmosphere is and the more intense the convection will be, to restore the lapse rate again. But these are meteorological deviations that can happen. We are talking of an atmosphere which is already in a stable state (has already a correct lapse rate), and about the convection in this stable state. We're not talking about momentarily and potentially large deviations from that state (which cannot last very long).