Possibly. Under some circumstances it could cause the battery to short out. If the engine has been running hot for long periods, the heat could damage the battery too. It might also be that the engine has internal damage and the starter is having a problem turning it over.
Yes it can. A faulty thermostat can cause overheating which will cause coolant to be pushed out of the system.
The system was overfilled and purged the excess coolant.
I'd recommend you do a battery test and charging system test. The battery would be my first suspect.
Low coolant? Cooling system airbound? Low coolant? Cooling system airbound?
No.
if you have antifreezecoolant in your oil you need to tow your car to get it fixed
There is probably a dead short in the system.
No, low coolant would cause the opposite.
a leak in a head gasket could cause coolant to go into only one cylinder.
Blocked radiator, low on coolant, stuck thermostat, air in cooling system
broken radiator
A bad alternator/generator would cause a 'no charge' in the system, which would drain the battery while running. Also a shor tin the wiring may cause a false indicator light.