I agree it wouldn't take much for them to find out. The important point is whether the manager feels there's any pressure on him to pick someone for non footballing reasons. If he's found out about wages on the side, that wouldn't put him under any pressure.
Nope.
Yes, it's certainly desirable. You want the manager to be free to pick the best team to do the job. It would be counter productive to do otherwise. And since it's desirable, it's obviously credible too.