Computing Current from known Capacitance and Time to Charge to Known Voltage
In general, the average current is equal to the change in charge divided by the time taken. But the change in charge is equal to the capacitance C multiplied by the change in voltage, which you know :-

$$I_{AVG}=\frac{C\cdot\Delta V}{t}$$

Your last paragraph is harder to address. The average current is not usually the same as the RMS current which will depend on the how the current varies with time. We would need to know how the charging/discharging takes place, for example constant current or through a constant resistance to a known voltage. If you are cycling the charge/discharge process, the long-term average current will of course be zero but the RMS current will be non-zero.