How long does it take for a battery to charge a capacitor?

Answer 1

Usually a tiny fraction of a second. Actually it will depend on the characteristics of the the capacitor, and of the remaining circuit (mainly, any resistor in series). The "time constant" of a capacitor with a resistor in series to charge from 0 to a fraction of (1 - 1/e), about 68%, of its final value. This time is the product of the resistance and the capacitance. After about 5 time constant, you can consider the capacitor completely loaded for all practical purposes - i.e., it will be at the same voltage as the battery.