Moris
...
Yes, as in communication between the two.
Whether it was willing or not, it states Ulfric proved his worth as an asset and was of use to the Thalmor. And seeing as how contact was established after the war, when Ulfric had escaped, I hardly see how he could unwillingly communicate with them unless he was blackmailed. Likely over the fact he was led to believe the information he gave up was crucial in the taking of the Imperial City.
I'm sure he knew who he was talking to seeing as how he broke off communication himself following the Markarth Incident.
There's no reason to believe that he knew who they were. You simply cannot demonstrate that.