So can someone help me with the DA:I Hawke timeline? The wiki is rather coy with the details and my memory is crap. So, pro-templar Hawke was Viscount in Kirkwall until the crazy red lyrium templars ran her out, whereas pro-mage Hawke split at the end of Act 3. But then... Hawke was investigating red lyrium and went to her Grey Warden contact about it, which is where she learns about the fake Calling. So... am I right in assuming that even pro-mage Hawke was investigating red lyrium, while she was on the lamb running around Thedas and evading pursuit by The Powers That Be? Or maybe that's just something she started recently once the red templars started crap in Kirkwall, whether or not she was Viscount? Help a fanfic writer out?? 
Yeah, as you said;
Pro-Mage Hawke was away from Kirkwall post-DA2 and at some point decided to investigate red lyrium, eventually going to the Wardens for answers because they knew more about the Deep Roads and possible leads on the Primeval Thaig than anyone else.
Pro-Templar Hawke was Viscount for some time post-DA2, until the Red Templars usurped them and forced them to flee, which is when they went to the Wardens for answers for the same reasons as above.
I suspect that Hawke was Viscount when the Red Templars overthrew them, probably sometime around the severing of the Nevarran Accords in 9:40 and long before the appearance of Red Lyrium and the Templar Order being compromised became apparent to anyone. It probably happened after Cullen left to join the Inquisition around that same time, since he'd have mentioned any coups or red lyrium use otherwise.
My pet theory on why Kirkwall was one of the first places co-opted by Red Templars, is that it was a targeted bit of revenge by Corypheus for brutally murdering him. He's got a major ego and a bit of a vindictive streak, so if he couldn't easily get to Hawke, I can see him taking it out on the city that Hawke was Champion/Viscount of, by sowing even more chaos there.