Does current flow stop when a capacitor is fully charged?
For a capacitor charge Q = capacitance C multiplied by voltage V.

Rate of change of charge is current and this leads to: -

\$\dfrac{dq}{dt} = C\dfrac{dv}{dt}\$ = current

This quite simply means that a rate of change of voltage gives rise to a current. If the voltage is rising linearly with time, the capacitor will take a constant current and once the voltage stops changing the current is zero.

In a real capacitor there will always be a small leakage current and this can be modelled by a resistor in parallel with the "perfect" capacitor.