The Alternator might be bad on it, and needs to be replaced, or at least looked at by someone who knows what their doing.
A blower motor that only works on high is normally caused by a defective Blower Motor Resistor Pack. DRL, do not have a clue.
Yes, a loose battery cable will prevent the alternator from charging the battery. The drain on the battery is not caused by the engine not running unless an accessory is left on. Most likely, the drain is caused by accessories (radio, lights, etc) while the engine is running due to the battery not receiving its charge from the alternator and then not having enough charge left to start the engine.
An Airbag service light can be reset by fixing the problem that caused it to go off. Unplug the battery for 30+ minutes, or clearing the codes with an OBD scanner.
You have 2 problems. First, the blower motor running only on the high speed is caused by a defective blower motor resistor pack. Replace it. Second the fact that it is only putting out hot air indicates the temperature control valve is defective or stuck.
I suspect that your outside unit is not running and that it is your blower only that is running. This could be caused by a sensor that actiaves the blower when the heat exchange is heated for the heating of your home. If it is warm enough and/or you have a standing pilot the sensor will make the blower run. Also, some units have a switch on them that can be set so that the blower runs all the time as well as the switch that your thermostat may have. One of the switchs may have been filp or bump to on.
This is caused by a broken heater matrix. I have had this problem and its is a real pain to fix as you need to take most of you dashboard off to get to it.
THERE IS A CCRM MUDUAL LOCATED NEER THE BATTERY IT HAS A RELAY FOR THE RIGHT FAN AND ONE FOR THE LEFT FAN. YOU HAVE TI REPLACE THE COMPLET CCRM UNIT.
The issue may be caused by a faulty blower motor resistor.
You might have caused the alternator to overcharge which could have damaged the computer at the worst, or blown a fuse at the least.
The bulging is caused by gassing. This is caused by overcharging or by a short circuit inside the battery. The acid inside the battery is turned to gas and it expands the sides of the battery.
Something must be draining the battery while it is not running. Usually it is caused by an interior light being left on either inside or perhaps the trunk or one of the doors were not closed completely and thus causing the interior lights to stay on and drain the battery. Otherwise, this will continue and you should plan on replacing the alternater and/or battery
a loose connection will do that if local near connector