No. Current is not stored. It is not stored in a battery, either. Current is a measurement of the movement of electrons to put it overly simple. The electrons have to be moving to have a current measurement.
Capacitors themselves offer opposition to current flow. This opposition arises from the fact that, at a given voltage and frequency, the number of electrons which go back and forth from plate to plate is limited by the capacitance of the capacitor (Farads). As the capacitance is increased, a greater number of electrons change plates every cycle, and (since current is a measure of the number of electrons passing a given point in a given time) the current is increased. The change in current that you have described is due to the change in capacitance value by series or parallel configuration. See your series and parallel formulas for capacitors to figure the capacitance value of capacitors connected in series or in parallel and this will be more clear. You are directly affecting the value of capacitance in farads.
Increasing the frequency will also decrease the opposition offered by a capacitor. This occurs because the number of electrons which the capacitor is capable of handling at a given voltage will change plates more often. As a result, more electrons will pass a given point in a given time (greater current flow). The opposition which a capacitor offers to ac (called capacitive reactance, Xc) is therefore inversely proportional to frequency and to capacitance.
MP