When batteries are connected in series for charging, the charging current is the same in each battery. If the charger supplied 1-Ampere of charging current to the series-connected two 12-Volt batteries, then each battery will get 1-Ampere of charging current passing through it.
To answer the questions several assumptions must be made:
- the battery charger only has two connections, a positive and negative, and cannot tell the voltage across individual batteries in the two-battery series string
- the charger determines the state of charge of the series connected string by the total voltage of the two batteries
- the charger's algorithm for apply charging current is based on the voltage across the two batteries
With those assumptions, I will propose these answers:
Q1: What happens if one battery is weak and takes a long time to charge or never gets to full-charge?
If one battery needs more charge than the other, its terminal voltage won't rise as fast. Since the charger only sees the total terminal voltage of two batteries in series, it won't see a terminal voltage that it thinks is indicative of full charge until both batteries get to full charge. It is also possible that the charger might see enough voltage to stop charging if the more-charged battery has an unusually high terminal voltage.
Q2: Will the charger keep going and possibly burn out the good battery?
The charger will continue to apply charging current until the total voltage of the two batteries reaches the threshold that the charger thinks is representative of full charge. This suggests that if one battery is already at full charge, it will continue to receive more charging current. To know if this will cause harm to the battery, we must know more about the battery. If the battery is a flooded-cell vented design, the over-charging will just tend to boil away electrolyte. If the battery has a sealed case and is designed not vent to the atmosphere unless there is dangerous pressure, the battery may or may not vent electrolyte, depending on what the release valve pressure is set for and the pressure build up.
Q3: Or, will the charger stop when the good battery is full-charge and leave than other one undercharged?
The charger will stop when the combined voltage reaches its threshold. If the batteries have unequal terminal voltages, the charging could stop when one battery is charged beyond full charge and has an unusually high terminal voltage, leaving the other battery with a lower terminal voltage which could mean lower state of charge.