I thought Templars were technically clerical, like the brothers who form the low ranks of the Chantry. They seem somewhat foggy on the particulars - sometimes, it's made clear Templars are supposed to be married to the church and their faith (that Templar in the Broken Circle who dreamed of a wife and children) and other times they marry and have children (like Aveline's husband).
I assumed as Chantry hierarchy, Templars were humans only - there are Dwarven and Elven Andrasteans, but the only Dwarven brother you ever meet is Brother Burkel. Some dialogue makes it clear that the chantry does not recognize Elven and Dwarven clerics. Sister Giselle admits that the official reason is that Elves and Dwarves "fell farther from the maker" but that the real reason is likely just politics and racism. I suspect either Brother Burkel wasn't recognized, or he was made a special case to preach in Orzammar.
The Last Flight's timeline is pretty foggy, which makes me wonder if the "canon" Divine is Leliana. She wants to ordain Elves, Dwarves, and Qunari, and I'd assume that would mean Templars as well. The books all pick a canon because taking into account all player decisions would mean the writers would have to avoid too many topics. Maybe the dwarven templar is there because of Leliana?
EDIT: For comparison, the real-life Knights Templar who were a religious order within the church were ordained monks as well as knights. They could not own property, could not marry, and could hold no debts. Except for the one married to Aveline, there's no sign that the Templars in Dragon Age are different. Maybe the marriage to Aveline was a dark secret? She didn't seem to care for chantry rules much. If the brothers aren't supposed to be non-human, I'm guessing that (pre-Leliana) the Templars who defend the faith weren't supposed to either.