Does the charge that flows into the capacitor during the charging process go all the way through the capacitor and back to the battery's or does it get stored somewhere in the capacitor?

Answer 1

A capacitor stores electrical charge. When the device is shut off, the charge will - over time - slowly bleed off. But immediately after shutdown, the charge is still present in the capacitor. In large capacitors, such as those found in air conditioning units, older style televisions, and microwave ovens, the stored charge can be strong enough to be potentially lethal. Service technicians have safe procedures for bleeding off the capacitor charge, but those not knowledgeable about electrical systems should not touch a chassis of above-mentioned items, even if shut off and unplugged. Stickers that say "Danger - High Voltage" mean exactly what they say.

Another Answer

A capacitor does NOT store charge; it stores energy.

The net charge on the plates of a capacitor is no different whether it is 'fully charged' or 'fully discharged'. It does not take charge from the supply and store it on its plates! It simply allows electrons from one plate to be transferred to the other, so there is no change in the overall amount of charge.

What it does do is to store energy, expressed in joules.