Well, first of all, the wire (the potentiometer) contributes to the potential difference: $\lambda =V/L$ where $L$ is the length of the wire and $V$ is the potential difference across the whole wire. $\lambda$ is the potential difference per unit length.
So, as you cover more and more length across the wire, you are effectively will get zero deflection on the galvanometer because the potential difference across that length of the wire would equal the voltage or emf of the cell that you were trying to measure.
I did not understand the second part of your question. The third bit:
Does current have to be flowing through a point before potential at that point can be determined?
If there is a potential difference across the wire, a current has to flow. It would do so if the wire remains intact. So, when you do measure the galvanometer deflections, it does so because there is a current flowing.
Galvanometers have soft iron core with wires wrapped around it and placed in a radial magnetic field. And there is a spring attached to the core with a pointer. When current flows through the wires, the magnetic interactions would produce a torque and that causes the pointer to move.