Laplace,
"So we can see electrons going into one wire of the capacitor, and we see electrons coming out of the other wire of the capacitor, but no current exists through the capacitor?"
Yes, that is correct.
"Is that because it is not the same electrons going in as are coming out?"
No, while the above statement is true, that has nothing to do with no current existing through a capacitor.
" But the same situation exists with just a wire, electrons go in and electrons come out and they are not the same electrons (because the drift velocity is low), yet we say that current flows through the wire."
Yes, current does exists through a wire or resistor, but the situation is different in a capacitor.
"If we have a branch circuit from point:A to point:B and a wire carries current from 'A' to one terminal of a capacitor while another wire carries current from the other capacitor terminal to 'B', then current flows from 'A' to 'B'. How does it not also flow through the capacitor?"
Because a capacitor accumulates and depletes a charge on its plates. It does that because its dielectric has a theoretical infinite resistance. If it did not have this infinite resistance, the capacitor action would not be possible because no charge would be able to accumulate efficiently. Because the resistance of the dielectric between the plates is infinite, no charge can pass through the capacitor and therefore the current through the capacitor is zero. The charge flow stops on one side of the capacitor where it accumulates. An equal amount of charge flow starts on the other plate on the opposite side of the capacitor. None of the charges go through the capacitor. Charges do not accumulate in a wire or resistor because the path is completely conductive.
Ratch