Referring to your first question - just to illustrate Art's answer I have attached a typical operating diagram (this one is from a packed column) which clearly shows Art's point that as the liquid flow changes there is a relatively small change in pressure drop, but large changes in pressure drop as the vapor flow changes. The dot in the middle of the diagram is the design point. The operating diagram for a trayed column will be similar.
Your second question is much more involved. The answer depends on where you are in the operating range of the column. At low flow rates you will actually find the opposite effect, i.e. that as the flow increases the efficiency of the column improves. This is because when the flows are low there is insufficient energy to ensure good mixing of the (separate) fluids, and the boundary layers will be relatively thick. Increased heat input, and therefore increased flow and mixing improves the situation.
However, when you are above the normal operating point you are approaching flooding. In this situation the rising gas physically carries (entrains) some of the liquid upwards, thus working against the expected separation of the components i.e. the liquid which is low in the volatile component moves up the column instead of downwards.
The usual behavior is to find the efficiency increasing with gas flow rate until a fairly broad plateau is reached, and then a rapid fall off in efficiency at very high rates as the column starts to flood.
You should read up on this sort of thing in a good book like Kister, Treybal, Perry or Coulson and Richardson.