If a tail dragger airplane is converted to one with conventional tricycle landing gears, how will this affect the balance of the airplane? It is possible to offset the effect by shifting weights around the airframe?
As a taildragger
And with tricycle gear (source)
Since the majority of the mass remains unmoved (the engine, fuel tanks etc) the CG will see a bit of a shift but will not be that drastic. The biggest weight shift will be moving the mains up front. However since your gear is located in a different place and thus is loaded differently your CG limits may change.
Not completely sure what you are asking. But in a tail dragger aircraft the center of gravity (CG) is behind the main land gear (MLG), while a tricycle configuration places the CG in front of the MLG. You achieve this by placement of the MLG, tail draggers generally have their MLG mounted near or forward of the lending edge of the wing, while tricycles' MLG are placed toward the middle or aft part of the wing.
Of course moving the landing gear around will move the empty CG of the aircraft and a tricycle configuration will generally weigh more.
There are many kit planes that allow you to build the aircraft either way, but I don't know any that would allow an easy conversion from one to the other.