Question on electric flux In the first diagram(given below), the charge should not feel any force as the electric fields are not touching it.

But now if we see the flux throw an imaginary cube, it will give out an electric field which means the point charge should feel a force.

Electric flux definition:- the measure of the electric field through a given surface.
I am in high school, we just started electrostatics and I am confused about electric flux please explain.
 A: 
consider a negative plate on the top right, now will the test charge
  move?

Yes. But not for the reason you think. In your first figure you have drawn electric field lines as if they exist only over a finite, for want of a better term, "width" and do not include your charge in question. In actuality the lines connecting the charges as you move from the center towards the ends of the plates spread out as I have attempted to show by overlaying some of them on your diagram below. 
If you take a close look at the link you provided, you will see that the electric field is uniform between the plates until you get to the edges where you have what is called the "edge effect". The effect is minimal as long as the separation between the plates is much much less than the dimensions of the plates, which is the case depicted in the link where it only shows one curved field line at the two edges to show this effect. This is what you have in the case of capacitors where the electric field is considered confined to the space between the capacitors. But the greater the separation, the greater the edge effect and the more field lines there will be outside the area between the plates. If the plates are located very far apart compared to their size, they will look like two point charges at a location between them.
I have taken your upper drawing and crudely (and not necessarily accurately) drawn in additional curved field lines connecting the positive and negative charges to give you the idea. Note that your charge is now within the field and would therefore experience a force and move. The force may be very weak depending on what the actual field strength is. You can see that the density of the lines gets less and less as you move from the center between the plates outwards. (The gradient would begin sooner the farther the separation between the plates.)
Your second diagram is incorrect. No electric field originates in the box to influence your charge. According to Gauss' law the net outward flux across a closed surface equals the total charge enclosed by the surface divided by the electrical permittivity of the space. Your figure shows no charge enclosed in the second box. All field lines entering the second box exit it as well so that there is no net outward flux across the surfaces of the box. You may be learning more about Gauss' law later.
One more thing about electric field lines. They are a useful drawing tool to provide information about both the direction of the electric field in space and the relative strength of the field. The arrows indicate the direction. The relative density of the lines in a particular region of space is an indicator of the relative strength of the field. In actuality the electric field exists everywhere in the space between the lines shown in the diagram. 
Hope this helps.

