Do filter capacitors always have a leftover charge when AC voltage is removed?
There is. Even a capacitor that has been immediately discharged will, over time release more charge from the dieelectric causing it to rise in voltage unless it is bled out. The effect is called dielectric absorption and for a capacitor operating at high voltage capacitors this can be significant.
I'm not sure if all types of capacitors do this, but the big electrolytic types do.