Would a 2nd row of distributed props on a 2nd staggered wing of a biplane increase its efficiency?
Source: nap.edu

Not really no. The velocity off a propeller does not shoot straight back, rather in circular motion (shown above and below).


Source: High-Lift Propeller System Configuration Selection for NASA's SCEPTOR Distributed Electric Propulsion Flight Demonstrator, NASA

From the conclusion of that NASA report on span-wise propellers generating lift:


  A potential constraint on the design of the high-lift propellers system is the total thrust. If large amounts of lift augmentation are required from the propellers, then high thrust values from the high-lift propellers are likely. If these propellers produce excess thrust, then the aircraft will be unable to sustain flight at the desired speed. This implies that drag producing devices may need to be added to the aircraft or the primary propellers operated as windmills to produce additional drag.


Applying the same conclusion to the biplane, thrust will be wasted, and possibly will work only in a limited angle-of-attack range for the prop-boosting-prop-boosting-wing idea.


Own work; variable pitch blade

Note that a very fast stream before a propeller leads to [forward] thrust losses due to the propeller's angle-of-attack (shown above). And the less efficient the thrust becomes, since the momentum change will be harder – propellers add little velocity to large amounts of air, in fact, the pressure rise (a thrust measurement, which also is indicative of the mass flow) is around only 2% for propellers.