What happens if an AC voltage is applied to a battery?
It's fairly common to see a lead-acid battery charged using rectified AC. As long as the charging current isn't beyond the capability of the battery, it will 'work'. If there isn't a series resistor somewhere, or some primary-side limiter, the winding resistance of the transformer could be what's limiting the charging current. 

A handheld multimeter is sensitive to 60Hz AC, so yes, your DC reading was likely skewed by the low-frequency ripple.

The circuit you have drawn is a full-wave rectifier.

If there isn't any explicit current-limiting protection in the charger, it is possible that the charger can become damaged if subjected to long-term overload. Wall-wart adapters have similar failure modes (usually the transformer goes high-impedance).