Therefore we will conclude that whenever a negative charge moves from
a lower potential to higher potential, work done by electric force is
That is correct. The electron gains electrical potential but loses electrical potential energy. The energy for doing positive work on the electron comes from the electrical potential energy of the charge field system. If the electron is unrestrained it will accelerate and gain kinetic energy equal to the loss of electrical potential energy.
...and when a positive charge moves from lower potential to higher
potential the work done by electric force is negative. Why? What is
Correct, but you are missing an important additional point.
An external (to the field) agent is needed exert a force to move a positive charge from lower potential to higher potential against the repulsive force of the electric field. Since the direction of the force exerted by the external agent is the same as the displacement of the charge, it does positive work. If the charge starts and ends at rest, the electric field does an equal amount of negative work since its force is in the opposite direction of the displacement. Assuming there are no dissipative forces involved, the net result is the negative work done by the field takes the energy supplied to the positive charge by the external agent and stores it as electrical potential energy.
The gravity analogy, in the absence of dissipative forces, is you do positive work lifting an object to rest at a height $h$ above the ground and gravity does an equal amount of negative work taking the energy you supplied and storing it as gravitational potential energy of $mgh$.
Hope this helps.