When Brit rides up to the Kiowa braves, he gives each of them a blanket. The next scene they are riding into the Indian camp and none of the braves have a blanket.
No military command would have built the fort where this fort was built. It is in a valley, looking up at a ridge-line all around (which would allow the enemy to easily fire down and into the fort) . Further, it is near a hill where a potential enemy (such as the Commanche and Kiowa in this film) could (and did) get close without exposing themselves to defensive fire. This is military engineering 101 and could (and often did) lead to a military disaster.
One of the Texas Rangers has a Remington Rolling Block. As the setting is 1861 and the Rolling Block was not invented until 1865, this rifle is a little out of place.
The Colonel tells Alan that he has just been ordered East to Fort Leavenworth. But Fort Leavenworth is North of Texas, NOT East.