Charges at rest move when a force is applied on them and this is due to Newton's laws.
Now to apply a force, we need a field, like electric/gravitational field. Each field acts upon certain measurable properties of a system, like gravitational on mass, electric on charge etc.
Now potential is just a fancy name of height in electromagnetism. I hope you're familiar with gravity in your day to day experiences, now, let's take a look more carefully as to how is gravity applying a force on you, it takes you from a higher height to a lower height on earth. This height is gravitational potential.
Similarly, there is an electrostatic potential, it tells you that electrostatic force will take you from a higher electrostatic height to a lower electrostatic height, but since electrostatic force isn't always attractive, it can take you from a lower height to a higher height as well.
So, first comes electrostatic force and then comes potential, kind of like without gravity, you don't know what is high and what is low. You associate tallness of something as to how much effort you'll have to apply to climb it up.
So, now the answer to your question.
$2$. The movement of charge is indeed due to an electric field, but no charge will experience a force due to its own electric field, so you feel a force only due to the other charge. This is just how we've found stuff to work till now.
$1$. Both the charges feel the force due to other one and will come to the centre, kind of if we both stand on a frictionless surface and if I pull you and you pull me at the same time, we both meet at the centre.
Also, you're used to thinking of height from a flat earth's perspective. For example, on earth actually, things that are on the same concentric sphere w.r.t. earth's centres are actually at same height, since it will take equal amount of work to reach all those points. So, think of height as how much work you'll have to do against the field force to reach that point. More the work, more the height.