I consulted with our "visa expert" Ken Hayes and this was his take on your situation:
Ken Hayes wrote:1. The FAO simply signed their names on contracts to expedite the process, calculating he'd save time since the foreigners didn't need to know about or be involved in the process.
2. The University is not licensed to hire foreigners. This would be unusual, inasmuch as legitimate universities are slam dunk assured they'll get a license. But, without knowing the history of the university, that is hard to determine.
The question is whether or not they have foreign expert certificates (FEC). They may not have been hired as "Foreign Experts" but "hired" as chemical engineers or the like via the Labor Bureau with the help of a company that has the ability to hire foreigners.
If they have FECs, then the school is probably legit, but should have issued them contract copies.
If the school is not legit, they could have used any number of ways to get the foreign residency permits.
If you have a foreign residency permit without having been given a signed contract, something isn't kosher. That much is for sure.
It's something you should be concerned about as a legal matter but, obviously, as a practical matter, all of you seem to think that there are no problems.