When King Stephen awakes from his nightmare in the dead of night, there is a full, roaring fire in an open fireplace. All fires would have been extinguished at bedtime and never, ever have been left unattended. The risk of the tapestries and wooden beams catching fire was too great.
The street music in Winchester that begins at 38:06 is the Trotto, an Italian piece composed 300 years after the period depicted.
The castles here and throughout the series are large, sprawling stone structures with battlements, moats, drawbridges and portcullises. In fact, most of the real castles depicted (Winchester, Lincoln, etc.) would have been simple wooden motte-and-bailey structures at the time. Round stone towers were unknown in England during Stephen's reign.
Elena refuses to marry Tom Builder because she sees marriage as the church dictating to them.
In actual fact, Medieval marriage customs did not require people to be married in church, or even in front of a priest.
They just had to exchange vows in front of witnesses. So Tom and Elena could easily have married without any involvement by the church.
Lincoln Castle is shown in a low-lying bowl surrounded by mountains. In fact, the castle is on the brow of a plateau surrounded by flatland.
Richard's ear was cut off in episode 2, but when his father is beheaded in episode 3, he hugs his sister afterwards and he has both ears.