If I were to build a J3 cub twice the size of the original using the same CG would it fly?
Unfortunately a simple scale-up would fall victim to the square-cube law: all dimensions times two means the volume and weight are 2x2x2 = 8 times the original, while the wing area is 2x2 = 4 times the original. It is the wing area that keeps the aeroplane up. It is the reason why birds cannot get any larger than they already are.

Someone in the 1920s analysed this and predicted that the maximum size for aeroplanes would be about that of the DC-3, but improved technology has proved that prediction wrong. A B777 has a way higher wing loading than a DC-3 of course. The square-cube law is not a law, it can be defeated by engineering. Point is that linearly amplifying all dimensions does not work.


  How big of an engine would I need, if the original is 65hp?


Minimal: 8 x 65hp = 520 hp. We need a serious amount of extra thrust.

From this answer:

$$D =  C_{D0} \cdot \frac{1}{2}  \rho  V^2 \cdot S + \frac{2  W^2}{\pi  A e \cdot  \rho  V^2 \cdot S} \cdot $$


$C_{D0}$ would stay the same if everything is scaled up
S = wing area = 4 times as high
W is 8 times as high.


$D_2 = D_1 \cdot (4 + 8^2/4) = D_1 \cdot 20$ 

So you need 20 times as much thrust at the same speed.

Increase V and the first factor increases, second factor decreases. There are of course many optimisations possible (weight saving, wing loading increase etc), but that goes beyond the scope of the question.


  What would be the weight of the up-scaled plane? Simply twice the weight?


Eight times that of the original. 


  What would be the stall speed?


Higher than that of the original because the wing loading is higher. Depends on the flaps &amp; slats installed.


  How fast would it go? 75mph?


Yeah it might..but it would probably need to go faster as @xxavier mentions.


  Is there a software program to test aircraft design?


FlightGear is an open source flight simulator that allows you to link your own flight dynamics model into it.