Can a capacitor charge and discharge at the same time?
No, the charge on a capacitor is increasing (charging), decreasing (discharging) or remaining the same.  There are no other possible states (assuming an ideal capacitor with no leakage).

When the capacitor is charging or discharging, there is a potential difference between the two terminals and apparent current flow.

This means a capacitor will appear to conduct an AC signal (above a critical frequency determined by the value of the capacitor and the impedance of the load/source) and will block a DC signal when used in series.