Does that violate the rule that electric field lines (at least those of the coulomb force) can only start/end at charges/infinity?
In the example you have given the answer is -no.
Your diagram shows the result of the addition of the electric fields due to two negative charges and it so happens that at one point the addition of the electric fields produces a net electric field is zero.
In between the two charges and along the line joining the two charges there are two electric fields, one from each charge and at one point on that line the net electric field is zero.
In light of the above case, how can we prove that a point of zero electric field (a null point) in an "arbitrary electrostatic field configuration" is necessarily a point of unstable equilibrium?
Consider the electric potential due to one negative charge, located at $(0,0)$, which is shown diagrammatically on the left hand side.
When another potential due to a negative charge, located at $(1,0)$, is added to the potential due to the charge at $(0,0)$ the potential diagram is as shown on the right hand side.
Consider a positive charge located at position $(0,0)$.
A small displacement along the x-axis will reduce its potential energy and it will feel no inclination to go back to position $(0.5,0)$ - unstable equilibrium.
However a small displacement along the y-axis will increase its potential energy and the charge will move back to position $(0.5,0)$ - stable equilibrium.
The position $(0.5,0)$ is called a saddle point.
There is no way of positioning the two charges so a not to hav a saddle point between them.
The reasoning can be done in terms of forces (equal to minus the potential gradient.
Moving away from position $(0.5,0)$ along the x-axis produces a net force on a positive charge away from the origin (unstable equilibrium) and movement along the y-axis will result in the positive charge have a force on it towards position $(0.5,0)$.