May God have mercy on you, from my experience with this car, electrical problems are a nightmare.
Lucky for you, I have experienced the same problem, and it took me 1 year to find the issue and fix it. This is what happened, after I've found one morning the battery dead, out of nowhere: jump started the car, and the AC quit working right then, plus I've got the airbag light on. One of the fuses inside the cabin is blown, as you will open the fuse cover on the side of the dashboard, is one of the 5 amp fuses on the top row (I don't remember which one). Don't replace it yet, as it will blow as soon as you turn the key on.
Open the hood of the car, on the passenger side you will see a harness that goes from the body of the car to the hood (it warms up the nozzles in the winter). Remove the insulation and you will see 4 wires, some of them/all are "cut" for the lack of a better word, due to the harness being too short (cheap ass VW). This happened due to repeatedly opening-closing the hood. The wires are touching each other creating a short, that gives you the AC and the airbag faults. How in the hell are all of them related, I don't know. Use some new wire to extend them and reconnect the broken wires (if any of them is still intact, might as well cut it and extend it now, while you are at it). Replace the blown fuse. Turn the car on, if the fuse doesn't blow again, you're good.
This will fix the problem, only thing left to do is find someone that has a VCDS and reset the airbags, clear the AC codes. The airbag code will not go away, unless you clear it. For the airbags, the error will be related to the front passenger seat, something related to the module that recognize it when the seat is occupied, for the AC, the error will be related to the G65 sensor. You will not have to do anything about those errors, besides clearing them with VCDS.
If you don't have anyone nearby, I will lend you mine for free, all you have to do is pay for shipping and a deposit that I will return to you once you'll send me the cable back.