Does the flap settings for the best lift/drag ratio change with speed on the Diamond DA20-A?
You did the right thing when asking a new question. Comments are poorly suited for the explanation of non-trivial issues.

Now follow several links to older answers. They should help to explain a couple of things, such as:


what flaps do to L/D.
how thrust changes over speed for propeller aircraft.
and how efficient propeller airplanes are in cruise.



  does the flap settings for the best lift/drag ratio change with speed?


Below you see the polar plot from the first linked answer. It shows the airfoil drag in 2D flow. Clearly, the best L/D is achieved with the +10° flap setting, but at this setting the airfoil becomes rather poor below a lift coefficient of 0.3. In order to reduce drag, the flap should be raised when flying fast enough so that the lift coefficient drops below that value. 



So the answer is yes, the optimum flap setting does change with speed. Only the flap setting for the overall best L/D will be found at the same flap setting, but then this best L/D can only be reached at one specific lift coefficient (which is equivalent with one specific speed if mass does not change).


  I can understand it's either up or down for the best L/D ratio but I don't get why does it change at a specific speed?


It does not change at a specific speed. It changes continuously with lift coefficient (and, implicitly, speed). If you compare specific flap angles, you will find crossover points when the L/D of one flap setting drops below that of another flap setting as the lift coefficient changes. Normally, airplanes only allow a few specified flap angles, so it might seem that L/D changes abruptly when changing flap angles. However, flaps could also move continuously, adapting the wing to its optimal setting for each operating point.


  why would someone design a wing that's less efficient to fly in Cruise then in Climb?


Cost. GA aircraft don't have the complex flap systems of airliners and, consequently, have a much larger wing than what is required for cruise. Wing area is sized for the mandated minimum airspeed of 61 kts or less and installed power means that the airplane at low level cruises at a speed where friction drag dominates and induced drag is almost negligible. While technically possible, the complex high-lift systems of airliner wings would make any GA airplane a lot more expensive, a death trap for untrained pilots and a maintenance nightmare.