Basic Capacitor Discharge
Current does flow through the capacitor, through the dielectric. But only while it's charging.
This discussion of capacitors may help. The charge buildup does repel further charge buildup. That's why the rate of current flow decreases as the capacitor charges. Work must be done on the charge to move it - the integral of that work is the energy stored in the capacitor.
Yes. The animation is very bad at showing this.
Yes, current will tend to flow so as to equalise the potential on both sides. Neither is necessarily related to ground. That's how a Cockroft-Walton multiplier works.