Hawke can be anything really...
Warrior fits the Champion aspect the most IMO.
But Rogue makes him the 3rd sibling (Bethany Mage, Carver Warrior) and fits the Kirkwall atmosphere the most IMO.
And yes, Mage does have the most exclusive story content/tone, but it also is damn weird in 1/2 the game how you're not taken in by authorities for being a Mage.
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Its pretty clear that there is no (or little) absolute canon for choices, but instead:
1)Default world state is just a sequence of choices that Bioware thinks is among the best to roll with, but only AMONG the best.
2)Bioware will try to accomidate every choice, but its never actually possible, so some choices get more coverage in future content than others. It doesn't actually mean there is a definite canon, or at least canon 'intent'.
I personally, however, feel pretty comfortable TRENDING towards SOME choices being 'more' subjectively canon. In terms of class (ALL IMO):
DAO - Very Warrior main plot. Very Human main plot. Therefore very Noble origin. Everything else secondary, but all get their highlights. Mage gets exclusive story and lore. Rogue gets lots of city and related content. Elf gets Brecilian Forest. Dwarf gets Orzammar. I want to clarify again that I don't equate default world state with what I consider more 'canon' to the single game itself. For all we know, any future Warden-related (not him appearing, but involving his future living story) could aim more to Elf lore, thus default world state of... Elf. but for DAO itself, it seems very Human.
For the record, I was a Male Human Mage who romanced Zevran.
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DAOA - So far, I'm going with Warden surviving somehow, doing Dark Ritual, but I'm not going to assert who did the Dark Ritual with Morrigan (though I'm leaning towards the Male Warden himself). I want to clarify AGAIN that I consider LOTS of choices just nearly as 'canon', and completely as valid.
For the record, I did not do the Dark Ritual with Morrigan, but I did have Alistair do it, so I survived.
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DA2 - Harder to tell. I'd say Act 1 leans more Rogue (searching for treasure), but Act 2 leans more Warrior (saving Kirkwall), and Act 3 leans more Mage (magical conflict) in terms of themes. Ultimately I'll say Warrior, but its even closer than in DAO. Hawke is simply Hawke. Hawke Hawke Hawke. Any choice you make just changes flavor - as in this is an even clearer aspect of the narrative than even DAO. Its very easy to consider Hawke a Mage due to marketing material and some events. Its very easy to consider Hawke a Rogue due to the extremely Rogue-ish setting you're mostly dealing with. Hawke is Hawke..
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DAI - This one is interesting! I'll divide stuff up into Haven (Lv 1-10ish), Skyhold (Lv 10-20ish), and what I consider for end/post-game (Lv 20+).
Haven - This is a super Chantry focused matter. You're the Herald, you have the Mark, you have been blessed by Andraste, you end a conflict between two warring sides, you suffer a siege, you mostly deal with humans. This is easy. Human Warrior. And even though there's no advanced classes yet, you could be considered Champion or Templar.
Skyhold - This continues the Chantry, but opens up to much more magical matters with the Fade, Tevinter, the nature of Wardens' Calling, political intrigue, assassinations, etc. Class-wise, I think things expand more into Mage and Rogue matters, but still a little more Mage. Race-wise, I think things go from being nearly totally about Humans, into being substantially about Elves, more than ever before. Tone and content and lore shifts. You're the Inquisitor (but a previous one was an Elf), you have the Anchor, you've been touched by the Fade, you search out ancient Elven truths.
End/Post-game - This depends. So far, we have the ending, endgame zones (Emprise, Wastes, etc), Jaws of Hakkon, and rumors of Deep Roads/Darkspawn and Fade/Travel/Qunari/etc DLCs. I can see both Human and Elven focus eroding into a more equal racial emphasis with Dwarves and Qunari. I can see a Warrior and Mage thing continue but be bolstered by at least some more content that feels more Rogue-ish.
Basically, I think DAI is more friendly (at different points) to 1/2 of the choices, and less friendly to the other 1/2 that were added later on or just have less story focus (Rogue, Dwarf, Qunari).
In terms of narrative, I think is goes Warrior --> Mage, and Human --> Elf. Rogues and Dwarves and Qunari are more along for the ride.
HOWEVER, if the supposed DLC rumors are true, we may get more for Rogues, we may get more Dwarf content, and more Qunari content. But its all DLC and not as much of a focus still so its more just to get these secondary roles more equal with the rest.
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Also, I think all of the games are completely fine with either gender as 'canonical', but I do think DAO gives somewhat more to Males, DA2 is generally equal or maybe a little more to Males, and DAI finally makes more of a clear flip and while its still super heavily Male, it is at least equal (in terms of its actual content, not marketing) with Females. All of them are practically equal, but there's some bits in each game that makes one go "Oh, they seemed to want a Male/Female a little more at this point, even if this works with either."
TLDR (~ = approx):
DAO - Male Human Warrior
DAA - Surviving Warden
DA2 - Male Hawke Warrior(~)
DAI - Male(~) Human/Elf Warrior/Mage
If this continues as some sort of trend, I wonder if we'll get a shift to:
-Even more female emphasis than DAI, to the point of having more marketing material than a male protagonist (even if only slightly)
-Finally a shift away from Human focus (even as it always continues; its Human after all) and not just into Elves (DAI) but a ton more for Dwarves and Qunari (assuming racial selection)
-Warrior story concepts from now on take a relative backseat (still big, like Human, but relative backseat) for more and more Magical Magery stuff. More magic than ever before, in various ways. While this happens, we also get an increase of things that continue a Rogue message, to maybe the point of a main game protagonist fitting the mould more than ever before.
I'm not actually against Female, non-Human, non-Warrior/Mages. I just think that DAO and DA2, and less so DAI, haven't emphasized them as much as the Male, Human, Warrior/Mages. But I think this is a thing that is ending, so yay, right?