Does the centre of charge behave as a point charge? We can find the centre of charge of a system of charges in much the same way we find the centre of mass of a system of masses
Suppose we have two charges of the same polarity and equal in magnitude that are placed at -x and +x along the x axis. The centre of charge would be at the origin.
Would this centre of charge, having magnitude twice as either of the individual charges, behave as the two other charges would combinedly? Like how gravity can be assumed to apply on the centre of mass?
If so, then can we say that for objects with a constant charge density that their charge can be assumed to be concentrated at their geometric centres?
If not, then what is the significance in defining a quantity as centre of charge?
 A: Answered:

... can we say that for objects with a constant charge density that their charge can be assumed to be concentrated at their geometric centres?

The answer is no.
A good rule of thumb in understanding these kind of things intuitively is "what you see is what you feel": imagine you are a little tiny you and can inhabit any point in the space. You have around you a spherical field of view, or "view sphere". If, in that field of view, you can "see" something charged that looks like a point (i.e. it occupies a small angular size thereupon), you will feel the force from a point charge, even if it's not really a point charge. If you "see" a plane of charge stretching to your horizon, you will feel force like a plane charge, even if it's not actually a plane. And so forth.
Now in your scenario, when you are near the center of charge of your situation, you don't see a charge there, so the effect of that center point is it has no effect: but you do "see" two charges on either side, and so you will feel influence from them as such.
Alternatively, one can think of this from the "third person" or test charge perspective: note that exactly at the center point between your two charges, the force on a test charge will be zero because the force from both exactly balance. Note also that this center point is the center of charge, if both are equal. Now consider a point just slightly off from it. By continuity, the force will be small, but not zero. In fact, for a time, it will grow as you move away from that center, and so away from the center of charge, which is emphatically not the behavior if you are talking about being close to a simple point charge. Hence the center of charge is not a point charge equal to the other two charges.
That said, if you are very far from the distribution, then you can, indeed, treat it as approximately a point charge: however, in this case, the distinction between "center of charge" and other "centers" also approximately disappears - you "see" both charges as being a single point in your field of view.
Also, I believe the above idea of "what you see is what you feel" can be made mathematically rigorous as actually exact, though I have not gone through the derivations to be sure, but it should be because the field around a charge looks just like the emanation of rays of light (as in geometric optics, not as in actual light waves in the EM field - think of the "field lines") therefrom, with intensity or surface brightness proportional to the charge. Hence the same mathematical rules that apply to the emanation and vision of light, will also apply to electric force (and also gravitational force, too, at least below the general-relativistic regime).
A: For an external point in the space, an electric charge uniformly distributed over the surface of an object behaves like a point charge of same magnitude located at the geometric center of that object. 
Example: A uniformly charged (metallic or non-metallic) sphere behaves like a point charge for all the external point under the consideration.  
The center of electric charges plays important role for an external point to find out the potential energy, electric field or electric potential.
A: No. For a simple intuitive example take a long rod containing a charge (uniformly distributed throughout its length). If you place a test charge  a small one that does not adversely affect the charge configuration on rod, it will obviously suffer more attraction/repulsion from the elemental charge closer to it than from one away from it. 
So, as you can observe, the behaviour of elemental charges charges about the centre of charge,  is not symmetric, and where there is no symmetry, such generalisations are not possible.

And as Mr. Harish Chandra Rajpoot said, the centre of charge can explain behaviour of an extended body in an electric field. Besides that, there is not much significance for the quantity
