Just from the symptoms, it looks like your battery is dead. Getting fully charged quickly and dying suddenly is a classic symptom of a dead battery. If the charger was at fault then it would never get charged to full. Just to make sure this is the problem, then you can try this experiment. Just get the battery full charged, stop the car but keep the ignition on and turn on your AC. A good battery should be able to run the AC for 6-8 minutes before it gets drained and engine restarts. If your battery gets drained in 1 to 2 minutes and engine starts then its a sure sign of a dead battery.
You loss of mileage is also because of a dead battery. Also check the battery filter and fan (under your back seat, which you ll be able to see after the seat is removed) and see if the fan is working and the filter is clean.
