If by "Does it escape that?" You mean "would a space elevator not be subject to the rocket equation?", then yes, it would not be subject to the rocket equation.

The rocket equation is about reaction mass that is carried by the thing expelling it. It does not apply to climbing a rope, nor does it apply to the flight of a helicopter.

Regarding pulling the satellite down,

Pulling down on the weight at the end is something we have to do anyway to keep it from flying out away from earth.