The simple answer: A capacitor will block DC between it's terminals, and pass AC.
If you connect a capacitor with one terminal on ground (earth), and the other terminal to a node that has both a DC and AC voltage, then the AC voltage will be passed through the capacitor to ground, but the DC voltage will remain.
As to answer the four possibilities you listed, I cannot say until I see a diagram of your circuits. (I often get parallel and series confused in my head, and I don't want to lead you astray.)
BEEF 41