Mechanical Property Responsible for Folding

I was wondering on which property does a material's tendency to fold depends upon. When we push a metal plate, it seems to move forward. However a rug would start to fold and form crests. A piece of paper moves forward but folds very easily when an obstacle is present in its path.
This tendency to fold seems to be dependent on dimensions of object as well. A very large metal plate would fold in presence of a 'push' force.
This tendency also seems primarily dependent on friction and any other component of force towards the opposite direction( opposite to direction of movement).
Can this tendency be quantified? If such a topic of discussion already exists then please share the link or name of book.

Yes. It's compressive failure. There are three main strengths considered in evaluating material strength; tensile (stretching), compressive (opposite of stretching), and shear (cutting).

What you are calling "folding" is usually called "buckling" or the sudden sideways deflection of a structure.

Think of your paper as a sideways column. Load is applied until it exceeds the compressive strength of the column, at which point the column buckles. Continued application of load will result in further deformations.

Elasticity has a lot to do with it.

Some materials respond with bigger displacements when stressed than others. In the case of a rug, it's a lot easier to make a rug bend than a piece of metal, because the rug is more elastic (that also has to do with it being a lot of thin fibres while metal would be a thick sheet that has to take all the stress internally). Paper is similar, it has a lot of thin fibres as well and therefore can easily bend (paper is also thin as a whole).

The thick plate can bend a little bit elastically, but to get it to fold like paper you have to do "plastic deformation" such that it wouldn't naturally want to go back to it's original shape.

The reason that a rug wants to bend easier than paper does have to do with friction. Note that friction force $F_f = \mu N$ where $\mu$ is the friction coefficient and $N$ is normal force. Normal force is just weight $N = mg$. The carpet has more weight, so the friction force is higher. This makes it easier to bend than to slide, the same as if you increase papers friction by weighing it down.