Why do we consider the electric field of an infinite plane? I never understood why one would calculate the electric field surrounding an infinite plane, if such thing does not exist. Is there physical motivation for using this model? Are the results applicable to real-world systems? Is it because of the mathematical simplicity?
 A: In physics and engineering, we often abstract and idealize a physical problem to gain insight into the physics, e.g., infinite plane of charge, infinite line of charge, point charge, etc.
Now, it goes without saying that if these idealizations didn't represent good approximations of relevant physical systems, they wouldn't be used.
With regards to your specific question, imagine a finite conductive plane with uniform electric charge density.  Far enough above (or below) the center of this plane of charge, the infinite plane approximation fails.
However, if the distance above the plane is small compared to the dimensions of the plane, the electric field is, to a good approximation, the electric field of an infinite plane and is exact in the limit as the distance above the plane goes to zero.
Another example is the ideal harmonic oscillator.  No physical system is an ideal harmonic oscillator yet, it's easy to show that many systems are, for displacements small enough, good approximations of the ideal harmonic oscillator.
Thus the value of studying and solving such ideal systems.
