Are Batteries Heavier When Charged?

Yes, the total mass of a battery increases when the battery is charged and decreases when it is discharged.

Do charged batteries weigh more than uncharged batteries?

We know that charged battery contains more energy than the dead one. Again this doesn't mean there is more mass or electron present in the battery, it just means there is more stuff present in the battery for gravity to pull on, which makes the battery weigh slightly more.

Why do batteries get heavier when charged?

The number of electrons in a battery is the same charged or discharged. To make the battery heavier, just take off the positive cable when charging. That's where all the electrons are leaking out.