Stay away from the non-mechanics at Auto Zone. They haven't graduated up to a battery load test YET.
the charging system is a bit complex. have it tested but you could have a failed alternator or a bad battery or even a blown fuse in either the supply or charge side of the alternator circuits.
Maybe yes maybe no. Never change parts with out getting them tested first. In many places you can either drive the car to the auto parts store and they will test them for you. If you can not drive it there take the part off and take it and they can do a bench test on them.
possibly but before spending the bucks be sure to verify the battery cables are in good condition having no corrosion inside the cable cords.
the only way to know is to check both parts. autozone will do this for free. a bad battery can damage a new alternator.
Could be. It also may be a bad cable or connection.