I'll be honest with you and tell you what CIC may see.
He married a woman to stay in Canada. She couldn't sponsor him, so he got divorced and found a new Canadian woman to marry to get sponsored.
Now, I am NOT saying that's accurate and I'm not doubting your relationship with him, but I do think the IO may interpret things this way. You should prepare to have an interview, which means a very long process. I'd rather someone warn you now rather than it come as a surprise. It's easier to prepare when you know ahead of time.
Do they have proof of their legal separation since 2011? You will need legal proof. (The application demands it. This isn't unique to you.)
He should definitely file Inland with the open work permit application if he wants to remain in Canada. Like I said before, generally CIC allows the spouse to remain in Canada until a decision is made. There is no guarantee and sometimes spouses are deported anyway, but it depends on what stage CIC is at with the foreign national. Just understand that filing a PR application will alert CIC to his overstay. There is no way around that unless he restores his status and that is highly unlikely in the given situation. CIC knows he wants to stay in Canada permanently.
It's impossible to have filed an extension in 2012 and not heard a reply. If he tells you nothing has happened, I'm afraid he isn't being honest with you. He was most likely rejected, is illegal, but doesn't want to tell you.