Is there a displacement at all when the resultant force equals zero and the body is at rest? Force is the change of momentum in time. But suppose a body is at rest on the floor. The system is in mechanical equilibrium, meaning that the forces acting on it sum up to zero and there is no momentum because there isn't any displacement.
But if there isn't any displacement at all, how can I say that these forces (each one being cancelled) do exist? Don't I need acceleration, and thereby velocity and displacement to observe it's existence? Does something like a microscopic displacement exist?
 A: A spring balance can be used to measure the forces acting at equilibrium. Strain gauges may also be used for the experiment. 
In your floor example you are in contact with the system and hence a part of it with no external agent in contact, hence you can't cause an acceleration.

But if there isn't displacement at all, how can I say that these forces (each one being cancelled) do exist?

The material(s) on which the forces are acting may respond by squeezing or expanding a bit and in some cases it may also be visible. 
In the study of motion, what matters are the properties of motion the object. If the object is at rest, it doesn't matter if forces are acting on the body or not. I mean, in both cases the net force is $0$.
A: Based only on a body at rest you cannot decide that force exists. That is correct. Obviously physics is based on many more situations. 
A: Do you agree that if you push on a wall you are exerting a force on wall even though neither you nor the wall accelerate? The reason is the net force acting on you and the wall is zero.
I should add, however, you can have displacement when the net force is zero. In the example of pushing against the wall the tissue of your hand can compress as well as the wall if it gives when you press against it. Those compressions can be considered as “displacements.
Bottom line: you can have forces without acceleration as long as the net force is zero.
Hope this helps 
A: Newton's Second Law of Motion states that the net force acting on an object is proportional to the acceleration of the object. F = ma. This is important because Newton's Second Law isn't related to a velocity, aka a change in position, but rather a change in velocity. So, for this reason if a ball is launched upward with an initial velocity v, it has a net force acting downward by gravity, but it is still moving upward. This can happen because that law relates the force to the change in velocity. So that change in velocity vector a.k.a acceleration will point downward. For this same reason, like you stated, for an object initially at rest on the floor with a net force of zero, the acceleration, or change in velocity will be zero. If you wanted to see if there were forces present in a system that is in mechanical equilibrium, then you could decipher that there were because gravity is a force that is always acting on us here on Earth. It pulls straight down, so if the object is at rest, and its velocity is not changing, then there must be another force present opposing that, the normal force from the floor provides this force to cancel out the gravitational force for a net force of zero. I hope this helps.
