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

The capacitor is used to store electric charge.That is what makes capacitors special. 
The charge that flows into the capacitor is stored on the plate of the capacitor that the source voltage is connected to. When current flows into a capacitor, the charges get “stuck” on the plates because they can’t get past the insulating dielectric. One plate is positively charged and the other negatively The stationary charges on these plates create an electric field. When charges group together on a capacitor like this, the cap is storing electric energy just as a battery might store chemical energy.

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