Are there normal forces in unbalanced forces

Normal force versus reaction force

Consider the following diagram:

The top block pushes the black box down with a normal reaction and the black box delivers the normal force (reaction) to the red box, so the net force on the red box becomes 0. These two forces (normal force due to the red box and normal force due to the black box) act on two different bodies, so that they cannot cancel each other out.

The red box pushes the black box through normal reaction, not its weight.

When the system is at rest, this means that the normal force due to the black box on the red box cancels the weight of the later box. This in turn means that the normal force between the boxes is equal to the weight of the red box.

It is not at all necessary that the normal force corresponds to the weight of the red box. There can be many other cases where such a system is accelerating (like an elevator) where the normal force is not equal to the weight of the red box.

When the system is at rest, the normal force due to the red box (the weight) increases to the weight of the black box. Now the bottom of the black box has to deliver the normal force that corresponds to the weight of both boxes if the system has to remain at rest.

For your final question, if your weight is more than what the floor can support it is actually going to break. Since you are in balance with the floor, the floor's normal response to you will cancel out your weight. However, you will break the floor if the normal response you give the floor goes beyond its limits.