You probably have mildew, mold or some variation growing on your coils inside your air handler. Best way to fix this if you have an AC / heater all in one unit is to turn on the heater for 10 minutes to burn off any of the growing molds and then pour some bleach down your condensation drip out pipe.
You probably have mold on the evaporator coil.
sounds like your air conditioner compressor is bad.
Bad smells in the aircon is usually caused by bacteria. You can buy special air con sprays that kill off moulds and bacterias making the car smell nice again.
Probably the evaporator core has mould on it and needs to be cleaned.
thermostat could be bad, or location of it, air flow over it.
It is likely germs and fungus in the air conditioner. You can use a spray product to clean it. If the evaporator is leaking you will get a very bad order. You will also start loosing your freon, then no a/c.
A bad alternator will cause a battery to go bad. It does not have an effect on the air conditioner.
The air conditioner or heater continually runs.
could be a bad bearing on the air conditioner clutch assembly.
You have a bad rear expansion valve located in the rear......
It sounds like the heater core is leaking. Check for the smell of antifreeze and that will confirm the bad heater core.
What most likely is happening is the A/C compressor has gone bad and its not able to turn when it is switched on. The burning rubber smell is probably the belt sliding across the compressor pulley that's not turning.