Since H1B or any other work visa and Green Card are parallel matters, I do not see why a person would need to work in the same company that is filing the I-140. I am pretty sure you can work at the University and get your I-140 processed through another company. Even though I-140 may be person specific, I do not think there is a requirement for the person to be an employee. 

I think so because, PERM does not give you a work permit, and by the time you file I-140, you still would not have a work permit. And as far as I know, H1B has nothing to do with green card..so what would happen to people not on H1B or other work visas, but still applying for a green card? Would they be working in the same company filing I-140? If so, on what legal grounds would they be working?

I am not sure about this, but I really wanted to put forth this argument.