They don't have to fight the same test, but at some point fighting is very likely involved. I think it highly unlikely that fighting is seldom involved in the Fade.
Nah. There are no actual fights in the Fade. This is the realm of dreams, where will turns anything into anything else. It doesn't work the same.
Also, there is an entry in WoT that describes a mage battling a demon for their harrowing - "The demon they made me face took the shape of a great cat. As we battled, it spoke in my mind. It told me that I would eventually stumble, and then it would pounce. [...]" Fighting seems standard.
Again, this is a battle of wills in the Fade. The spirit of Justice, Sloth, and even Mouse make this abundantly clear. They aren't actually combatting anything.
You said mages as a whole lack the raw power to combat demons - you did not say they weren't powerhouses. So, based on what?
Because for a mage they're the same thing. A fight in the Fade isn't the same as a battle in Thedas, yet. I'm not going to pretend that mages aren't powerful. They obviously are, but demons are stronger, faster, and more powerful than they, the higher ups anyway. The Templars have resistances to these magics, as demons and mages wield the same type of magic.
And mages aren't well trained after passing the harrowing? They're only considered full mages of the circle once they pass the harrowing - and they continuing training and studying long after the test. And templars can't be terrible soldiers? Okay.
Uh, no they can't be a terrible soldiers. They're charged with guarding the largest threat in Thedas and must stay vigil at all times over their charges. There's a screening process. And while mages can train, they will never go past their own natural ability without augmentation, lyrium being extremely dangerous, and blood magic. Neither of which are useful in this situation when compared with Templar might. Templars on the other hand are already selected to be above average soldiers and have already proven themselves as a whole multiple times throughout Thedosian history.