What is the physical nature of a "force"? This has application to E-M as well as gravity, but let me use the gravitational example as I think it's a bit more conceptually easy to grasp.
Say we have two 1kg weights placed on the ground a metre apart. We have a formula for the gravitational force they exert on one another yet on an ordinary surface they are unlikely to accelerate towards one another because the frictional force exactly counteracts the gravitational force.
So two questions follow: if this frictional force is exactly equivalent to the gravitational force does it have a physical nature like gravity (eg we now know that GR's postulations about gravitational waves are correct so there is presumably some sort of radiation between the two 1kg blocks but we surely don't postulate a radiation of friction).
Secondly, and relatedly: the 1kg blocks exchange "gravitational radiation/interaction" but this has no impact (as the friction negates it). So in what sense can we say this force/interaction actually exists?
Apologies for the rather naïve nature of the questions.
 A: 
the $1kg$ blocks exchange "gravitational radiation/interaction" but this has no impact (as the friction negates it). So in what sense can we say this force/interaction actually exists?

We are confident that the force exists because in every case where bodies exert gravitational forces on each other that are not cancelled, we observe the resultant acceleration. And in the situation you describe, all our experience of surfaces in contact tells us that there will be a frictional force. This argument fails, of course, if you insist (committing a category mistake, in my opinion) on defining a force as the product of mass and acceleration, or as rate of change of momentum.

if this frictional force is exactly equivalent to the gravitational force does it have a physical nature like gravity"

The frictional force may be equal and opposite to the gravitational force in the situation you describe, but it is hardly 'equivalent'. For one thing, the frictional force doesn't act directly between the bodies, but between each body and the ground. It also depends on the bodies being in contact with the ground. And, as already pointed out, it is essentially an electromagnetic force.
A: Force is the derivative of potential energy. If you have zero resultant force, you are in a minimum of total potential energy. Minimization, or more generally optimization, is most often the result of conflicting goals.
As an everyday example, think about how much you are willing to work. If you don't work at all, you don't get money for buying food, luxury goods, etc. But if you work all day long, you have no time to spend the money you earned. So there will probably be some individual way to resolve this goal conflict for you, where you earn just enough money to be able to spend it to your satisfaction. Your consumption will compensate the work's effort somehow, but it surely does not make much sense to claim that working plus consuming is the same as not working at all. Any deviation from the optimum will impair one or the other of your goals.
With forces it's the same. There are different kinds of forces, each with their own type of potential energy. If they act at the same time, there is an optimizational goal conflict, which gets resolved by the system finding a new equilibrium of the sum of potential energies. It does not make sense to deny the existence of forces just because different forces compensate each other under specific circumstances. It is more reasonable to assume that the forces are still there even in equilibrium, because if you deviate a little bit from equilibrium, you will notice the difference (one force becomes a little stronger than the other).
What happens to your two 1kg weights when you release them, is that they actually will accelerate towards each other due to gravitational attraction, but this will evoke friction (which is invariably connected to elastic forces in the involved solids) and hence, the frictional (electromagnetic) forces and the gravitational force will achieve a new equilibrium (after some period of damped vibrations), where the two masses are a tiny little bit closer to each other than when you released them.
You can test the existence of both forces by allowing a deviation from equilibrium. If you push one of the weights with your fingers (manipulate the electromagnetic forces), you will notice that friction keeps pushing back until a new equilibrium is attained. And if you load another 1kg on each of the existing ones (manipulate the gravitational forces), you will notice that they move yet another tiny little distance towards each other until friction again compensates the bigger gravitation in a new equilibrium position.
The idea that the deviations from equilibrium characterize the dynamics of a system is a very important way of formulating physics in general, which can even be generalized to when things are not static anymore but actually start moving. It is called the principle of least action.
