You can't discard that body from your calculations, even in a frictionless case.
To convince yourself of that you can just think of the fact that the box would accelerate without the need of any force, which is obviously absurd.
You are not considering the rope's tension, or you're not considering the fact that it's influenced both by the first and the second body.
You have to write down Newton's second law separately fro the two bodies keeping Tension into account.
$mg-T=ma$ for the "falling body"
$m_2a=T$ for the box.
You keep talking of friction but friction isn't a "requisite" for the box to slow down the falling object. In this case the falling object is slowed because it's pulling the box which has an INERTIA (a mass), and this means we have to exert a force to move it.
In this system you have the two object are accelerating at the same rate, right(because of the inextendibility of the rope)? So there MUST be a force responsible for the acceleration of the box and it is the tension of the rope. The force that keeps in tension the rope is given by the falling object wich "spends" a part of his acceleration to move the box. Try to understand the couple of equation I gave you and you will understand my argument.