I wonder if Cullen was in on Justinia's back up plan for the Inquisition. I can see Leliana and Cass being told by her, but I wonder if Cullen was told due to him being a high level Templar still probably loyal to the Divine.
I'd say it's very likely at this point.
I honestly don't know if he knew of the back up plan, but considering Cassandra is the one behind the formation and likely the initial bulk of our forces are from the former Chantry ones, and that we have no choice in the advisors, there is no reason Cullen got picked if he rebelled with Lambert.
About the 'factions', I think that we should keep in mind that they're not likely three indipendant organizations, but something inside the Inquisition (one already called a division of it, and Cullen seems to be the general of the Inquisition's forces. I doubt our army would be formed by another organization). We have at least one (most likely two) leaders coming from Chantry ranks. They didn't form indipendant organization prior the DAI events. . They aren't 'willing to work with you' in the sense that the joined the Inqusition with their forces. They probably joined it (in my opinion at its foundation) and get appointed as leaders of the various fields.
And I want to say that I'm VERY glad they dropped those stupid names

.