It doesnt add thrust. It decreases the losses from pressure at exit being lower that the atmospheric pressures at low altitudes.
Indeed. Personally I always liked the idea of “louvres” or “flaps” at the right diameter of the nozzle that would open (on the S71 I think they were referred to as “suck in doors” but at a much larger scale) when the pressure was below atmospheric. This way the engine is sucking in air from in front of (or above if vertical) so sucking the engine to the pad become pulling it off the pad.
BTW there are some NASA studies on (roughly) wrapping a rocket engine in a duct (or a duct around the stage skirt) that reckoned you could increase T/O thrust somewhere between 15-50% over a part of the flight. The joker is how heavy does that ducting have to be?
I never really understood why NASA didn’t pursue it. Their massive phobia about averting risk? Lack of a current mega-project to use it on? Who knows.
Breaking a flight into segments shows that takeoff is the time you want maximum thrust (you could say maximum momentum transfer between the vehicle and the pad)
Under the “simple, but crazy” notions in this area include (for example) a long umbilical to feed a water spray in the low pressure area of the nozzle in the first say 10m of takeoff. Just pumping extra reaction mass out the back. Then the connector separates and the reel retracts. I’m pretty sure someone (several someones probably) has already had this notion.
Like a lot of these BOTE notions however while the idea is simple the implementation is much more involved. 
As always does the performance boost you get in this segment justify the hit you take on complexity and mass?