At the start of the film, The Bruce hands over his sword and is searched before entering a church for a meeting where he has to defend himself without his sword. But when he walks into the church there are two shots where he clearly has his sword with him again in the scabbard.
Swords would never be carried just stuffed in a belt, they would be in a scabbard. And no one would thrust their sword in to snow or the ground for convenience. It's not good for the blade. Swords were expensive and your life depended on it.
Robert the Bruce tells Scot to "Go peel some potatoes for your mother." The potato wasn't introduced to Europe until the mid 16th century. The events of the film are in the early 14th century.
At about 58:00 minutes, Morag (Anna Hutchison) uses a knife blade to cauterize Robert's (Angus Macfadyen) wound. The tip of the metal blade is glowing red hot but she pushes down on the blade just below the glowing section. If even part of the blade was red hot, the entire blade would be far too hot to touch.
When Morag cauterizes the wound on the king's leg, she herself is pressing down on the blade with her finger which of course would cause her searing pain as well.