Is Varric's Tale of Champion supposed to be fiction or non-fiction? I thought Varric is known for stretching the truth, so why would the Inquisitor accept what is written by Varric as being totally accurate?
I think it was supposed to be and taken as non-fiction. They kind of teased Varric being an unreliable narrator in DA2, but seem to have decided not to follow through for DAI.
Which is a shame because, personally, I think this was a wasted opportunity for the devs to address or retcon issues the fans had with the previous game. They could have just said, "Varric's an unreliable narrator! This is how it really happened!" But no, they have him stand by everything he said, or have other characters confirm what he said.
Which is I think part of the reason the Inquisitor could be excused for believing him. Despite claiming to "tend toward extravagant lies" in Haven, Varric never really lies during the game, and/or his claims are confirmed to be true by other characters (including Cole)! Plus, depending on when the Inquisitor reads his book and confronts Cullen, she could have been traveling with Varric for months and/or befriended him. Why wouldn't she trust a friend?
Is there any evidence to suggest abuses were taking place regularly at the Gallows? It was clear in DA2 that the place was very much like a prison - basic freedoms denied - but was rape, beatings, and illegal tranquility the norm?
Other than its reputation for being a complete and utter hellhole, and Cole confirming that most Circles, Kirkwall included, are like that? 
Given Cullen's moral compass and his traumatic experiences I could see him supporting locking Mages up, denying them freedoms - but I cant see him turning his back on rape, beatings, and ill supported tranquility... am I off-base here?
This... is still a contentious issue among many fans. To put it mildly, some fans feel that Cullen being depicted as a better guy each subsequent game (kinder in DA2 compared to "deranged serial killer Cullen" of the DAO end slides; nobler in DAI compared to "you can't reason with mages; they aren't people like you and I" Cullen of DA2) comes across as clumsy retcons to the previous games. The DAI YMMV TV Tropes page even says that, to some, Cullen's complicity in Meredith's past crimes was either insufficiently addressed or clumsily retconned.
I wish it had been better addressed in DAI too, but... well, there's no taking it back.
What role did Cullen have at the Gallows? Was he mostly the one training the recruits? He seemed to spend his days outside the Gallows. Didnt he imply at one point he was not privy to all the Meredith was up to?
From what I understand, he was something of Meredith's right hand; or at least one of her higher-ups. And Cullen half-heartedly mentioning at one point that he didn't know what she was up to--see above.
Cassandra indicates in the Inquisition that Seekers knew that things at the Kirkwall Circle were not great, that there were many rumours, but that they chose not to investigate? Do we know why?
My completely objective and unbiased guess (
) is that the Seekers were biased in the Templars' favor.
Were the people of Kirkwall aware of the number of Demons/Abominations etc running around in the city? Like the guy that murdered those women and Hawkes mother or that Elf who killed his wife?
My guess is no, otherwise there would have been widespread panic. Didn't Meredith hire Hawke to track down the runaway mages to keep the public from finding out? Either that or the public did know how many Demons/Abominations were running around the city, and that's exactly why they, the Seekers, and every other non-mage supported Meredith's extreme measures.