Why does it make a difference if I consider wedge and block as two systems or one? In this setup:

I want to find out if the wedge will accelerate if both the wedge and block are taken as one single system. If the free body diagram of the wedge is drawn, the net force on it comes out to be zero. But if both bodies are considered as one system, then the system should accelerate due to the external force F. Why the ambiguity?
 A: Your calculations show the system accelerates because it does!  If you plot the center of mass of the system over time, you will see that it moves to the left, as expected.
If you treat the two bodies separately, you get a bit more information.  You see that the center of mass of the system moves a little to the left because the block (and its center of mass) move to the left quite a lot, while the wedge (and it's center of mass) do not move at all.
What causes confusion is if you do the calculations on the whole system, and then assume that because the center of mass of the system moves, everything in the system must move as well (including the wedge).  The property of "if the center of mass moves, everything moves equally" is a property of rigid bodies.  The system containing the block and wedge is not a rigid body, so thus this rule not apply.  The parts of the block+wedge system are permitted to move separately, and do so.
You could also play a game where you break the wedge up into two halves, just like you broke the system up into a block and a wedge.  If you do so, you will find the centers of mass of each piece do move together.  Why?  The wedge is a rigid body, so if you divide it up, you'll find the centers of mass of all the pieces all move equally.
The way you break up the problem does not change the solution.  Physics is physics.  It does, however, change what aspects of the solution are emphasized by the mathematics.  Choosing the "right" way to divide up a system makes it easier to generate numbers which you can apply practically.
