The nuns at the Abbey where Cecilia is sent are wearing the Brown Scapular often associated with Carmelites. While the Carmelite order existed during this time period, the Scapular did not. Its use began about 1251: 65 years after the end of the film.
Arn's trial by the archbishop is highly anachronistic in many ways. First, the church did not have any say in matters of marriage at the time the events are supposed to happen. Such were a completely private matter between the families involved until at least early renaissance. A daughter becoming pregnant outside of marriage would be the subject of her father's wrath rather than that of the church. Second, a bishop passing sentence like that would be a political matter which would not go without debate as to whether it was under the jurisdiction of the church at all. Being a noble, Arn would be the subject of judgment by peers. Third, the sentence in itself is absurd, as the knights templar were not in any way under the command of the church (nor was its members subject to mundane law). The organization was not aimed at harboring a condemned criminal, on the contrary, joining the order would require donations, arms, servants and the trust of its other members (and a solemn ceremony and vow). Fourth, the cloister where Cecilia is ushered to has a lot more in common with 20th century orphanages than with any medieval monastery. Sending a girl into a monastic order would be the decision of the family, not a sentence made by a bishop. Fifth, with most of the members of monastic orders being either people joining voluntarily out of a spiritual call or nobles sent there by their family for education, physical punishment would not be issued as depicted.
When Arns mother is about to die, he walks outside the monastery and sees a man practicing with a bow. At a distance, you can see the man shoot his third arrow. But when closing up, he shots again, this time it is also his third arrow (in both shots you can see two (2) arrows stuck in the target).
The cross worn by the Knights Templar (red border, white inside) is the cross of the Order of Christ, the new order formed by the Portuguese Templars made after the Knights Templar had been dissolved. The Templar cross was solid red; the shape of it varies on historical images.