Jamie putting on the red coat was not a part of any book in Outlander. Sam Heughan fought for the scene when Jamie puts it on. He said it represented everything Jamie had fought against and he had a very physical reaction to putting it on but used it to show Jamie's reaction to having to wear it.
The Battle of Alamance took place on May 16, 1771. Because Jamie had his birthday the day before, that means Jamie Fraser's birthday is May 15, 1721. In the Outlander novels, Jamie Fraser's birthday is actually May 1, 1721.
The British soldiers wear their hats with the front point straight ahead in the battle. Infantry troops actually wore their hats with the front pointed over their right eye so their muskets would not collide with the corner of their hats when they shouldered arms.
While in earlier episodes, Briana and Roger's son had dark brown hair like Roger, he now appears with blonde hair, making it appear that he is the biological son of Bonnet, who also has blonde hair.
Losses for both sides in the Battle of Alamanc, are disputed. Tryon reported 9 dead and 61 wounded among the militia. Other historians indicate much greater numbers, between 15 and 27 killed. Both sides counted 9 dead among the Regulators and from dozens to approximately two-hundred wounded. Tryon took 13 prisoners.