Historical howlers
In a fantasy context (which this show is, essentially), some leeway is allowable, but there were serious historical howlers in #1:
1. The Templars were not suppressed for being "too radical". They were suppressed because a) Philippe IV wanted their assets; b) they didn't want to be amalgamated with the Hospitallers. The true story is horrific and heartbreaking, involving innocent men being subjected to appalling tortures. I suggest reading Malcolm Barber's The Trial of the Templars, and Helen Nicholson's The Knights Templar: A New History.
2. It would have been pointless for any to flee to England (or indeed, Scotland), because there were arrests and trials in London (and at the Abbey of Holyrood), although not many, because there simply weren't many of them here.
3. The Templars were never linked to the Grandmont order, but originally had some links with the Cistercians. The Cistercian Bernard of Clairvaux was an early propagandist for the order.
4. It was said that they were given to "burning and killing thousands of Muslims" with the implication that these were peaceful innocents. The Templars fought Saracen armies in battle; I cannot think of any civilian massacres ascribed to them – although a few hundred of them were slaughtered when POWs after Hattin in 1187, and the Church which they had served tortured to death and burned a number of them in the early 14C. The episode fudges this, and the issue of Islamic military expansion in the Eastern Mediterranean. None of the major religions has spread peacefully, and violent imperial expansion has never been the sole prerogative of people of any one creed or skin-colour: it's dishonest to pretend otherwise for reasons of post-imperial guilt-tripping.
5. It was pointless for the Templars' assailants to dress as Turks, because the Templars were wanted men. Why dress up to attack them?
6. Friday 13th being unlucky is a modern superstition, 1880s-1900s, possibly US, and has nothing to do with the Templars.
7. Dovecotes, or doocots as we call them here in Scotland, were to provide fresh meat during the winter: castles, tower-houses, substantial farms, as well as religious houses had them. The pigeons were not symbolic, they were for roasting or putting in pies.
Seingner Conrat, tot per vostr'amor chan
http://www.silverwhistle.co.uk/knightlife