How is there motion in a horizontal spring-block system? Let’s say we have a horizontal spring system with a spring attached to a wall, and a mass attached to the spring.  If I pull the mass back, stretching the spring out, and then let go, the mass will accelerate, and the spring will begin to compress. I understand that as common sense.
According to newton's third law, if the spring is pushing on the mass with a force of for example $5N$, then wouldn’t the mass also be pulling on the spring with a force of $5N$? If the spring is exerting $5N$ of force onto the mass, and the spring is being pulled with $5N$ of force by the mass, why would there be any displacement at all?
 A: 
In this illustration one can see the spring system.

Newton's third law: All forces in the universe occur in equal but oppositely directed pairs.

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A system cannot "bootstrap" itself into motion with purely internal forces - to achieve a net force and an acceleration, it must interact with an object external to itself.

You are treating the problem as internal forces, whereas there is an external force imposed in stretching the spring, F1 at time T1, and the energy is transformed to potential energy. In addition to the internal equal and opposite forces sticking the spring to the ball, (Newton's third law) there are the additive forces of the atoms and molecules, dp/dt of the string giving up potential energy and pulling the ball in adding up to a force F2 at time t2.
A: The two forces associated with Newton's 3rd Law are acting on different objects, not the same object, so the net force on each object is not zero.
A: The spring is attached to the wall, and there is a force on the spring by the wall which in your case if the spring is massless, is equal to 5N!
You can see that the block is acted upon by a force of 5N by the spring and hence it moves, and the spring doesn't have a net force acting on it (as the force by the block cancels out the force by the wall) but still can move because it is massless.
If the spring wasn't massless, the force exerted by the wall on the spring would be greater than 5N and such that $(F-5N)=ma$ where $m$ is the mass of the spring and a is the acceleration of the center of mass of the spring.
