Can they buy the documentation of the employer records? Can they buy the documentation of the SS records?
Most likely they can do what ever it takes, not long ago an illegal managed to pass a federal background check, got hired as a US Border Patrol officer and was only caught when his fellow officers reported they believed he was smuggling illegals into the US. He was investigated, and only in the course of that investigation was it found he was himself an illegal and had used false documents.
I am sure not many are as clever as he was, but you never know, there is a well organized criminal system that thrives on false documents for illegals and they have been caught bribing various officials to get driver's licenses and other paperwork they need that can be cross checked.
Illegal entry into this country is organized crime in every way, so I will not feel comfortable that they could not accomplish matching records by some means. The other issue is the bleeding heart syndrome, if many are rejected because they can't provide the proper paperwork, the rules will be relaxed for them, open door for the criminals to walk in.
Yes and Yes. It happened in the 80's it will happen now.