Deliberate mistake: In the fight scene with the chariots Maximus takes one of the horses from a chariot and jumps on it. The horse already has a saddle complete with stirrups despite the fact it was pulling a chariot and not carrying a rider. (01:23:30)
Deliberate mistake: The historically inaccurate portrayal of stirrups in the film was actually due to the stuntmen enacting the cavalry charge in the initial battle scene. Apparently the stuntmen refused to ride the horses over the forest terrain at full gallop without stirrups because it was far too dangerous for both horse and rider. (Galloping a horse through forest or any uneven terrain is incredibly dangerous.) Stirrups were granted as a concession to the stuntmen on the grounds of saftey. Since they were included in this scene, the film makers decided to allow them in all other scenes where necessary. (Only seven stuntmen were used in the filming of these scenes - all the other riders are just CGI copies.)
Deliberate mistake: When the chariot crashes through the gate we see an explosion coming from both sides of the gate. However there is no reasonable explanation why there would be an explosion, so this was probably done to make the scene look more dramatic. (01:22:55)
Chosen answer: No, they are full brother and sister. They both had the same mother and father. Incest was not exactly normal at that time, but it stretches back within the imperial families as far as Caligula and possibly earlier. There is no evidence to suggest that the real Commodus was attracted to his sister, it was probably just included in the film in order to make the character seem more disturbed, and also as another reason why he would be Maximus' enemy (Maximus and Lucilla were, after all, once lovers).
The Doctor