I’ve heard cars whose airbags have deployed (for any reason) must be “totalled.” True.
Totaled just means the cost of repairs is more than the car is worth. Replacing airbags is expensive. Depends on the car.
I thought it may have had something to do with liability issues relating to the re-installing/resetting of the airbag mechanism. Who does it? Are they factory-authorized to do so?
Replacing popped airbags costs a few thousand dollars. It typically takes significant front end damage to trigger the airbag sensors which also costs a few thousand dollars to repair. On an average vehicle more than a few years old with higher mileage worth less than $10,000 it’s more cost effective for the insurer and provides better results to the insured to total the car and sell the car as salvage.
It’s not a solid rule; cars can be totaled without the airbags deploying and airbags can deploy without totaling a car, but it’s a pretty good guideline.
Yep. 
One big issue with airbag deployment accidents is that these days they show up on CarFax and other vehicle history services. That can make the car very difficult to sell even if it was a relatively minor accident and was properly repaired. If you’re in a situation where the insurance company is taking the diminished value of the vehicle into account when deciding whether to total it, there’s a very good chance an airbag deployment will tip the balance if it’s close.
If every airbag deployment resulted in a total loss my dealership would sell a LOT more new cars each month.
My parts department sells several airbags every week. They would not be doing that if the cars were a total.
Depending on the car model, year, and exactly which bags deployed a deployment can add between, oh say $1,200 and (pulling number from my ass) $20,000. 20K would be if everything blew which I would only expect in a rollover which quite likely would be a total before adding in the cost of the bags.
Is there still a big problem with counterfeits? (Not at your dealership, of course.)
I’ve never heard of a large issue of counterfeits, but I have seen warning lights removed and bag covers glued back into place making the unwary think the system is functional.
Scary.
I blew the airbags in a vehicle without even scratching it. I wasn’t totaled and I still drive it.
My impression was that an insurance company might decide to replace the air bags if it’s cost effective, but that practically no dealer will take the car as a trade in.
Any truth to that?
No truth to it I’m aware of.
Here’s some information:
Edmunds: Is Your Replacement Airbag a Dangerous Counterfeit?
ICE: Washington state man pleads guilty to trafficking in counterfeit air bags
It’s very fitting that you drove yourself and weren’t totaled 
I did get black eyes from the airbags, but I wasn’t totaled. 