Artillery are seen to be firing explosive shells. Shells did not come into common use until the late 18th Century whereas the film is set in the 17th Century. Artillery in the this period would have been firing solid cannon balls into masses of troops.
It is Ostap and not Andrei that warns Taras not to make fun of them and their clothes after they return home. Ostap is the more adventurous, whereas Andei's greatest characteristic is the deeply romantic feelings of an introvert.
When the brothers are sitting by the river observing Natalia on the opposite bank, in the front shot they are close together and the rear shot they are apart. Then the front shot reveals them close together again.
Lighting in some scenes changes depending upon the angle of photography e.g the scene with the brothers and Natalya by the river (the sunlight and shadows vary), the scene where the brothers return home (the Cossack watching the fight is in bright light in close shots and shaded when he is the background to the action).
When the horses are leaping the gorge in the challenge, the shot from beneath shows the horses crossing the gap with their legs dangling down as they are being lifted across rather than leaping across.
The cossacks' "scalp-lock" is not on the back but the front! Only the old cossack is the one that has it correctly. Also Yul Brynner's scalp-lock is very unnatural. It looks like a pony tail and it has too much hair for being only the remaining hair from shaving the rest of the head.
The Polish cavalry pursues the Cossacks from Dubno and is defeated, giving victory to the Cossacks. However, there is no mention of the infantry troops that are seen when the Cossacks first arrive at the fortress nor is the artillery used again when the Cossacks return. There would still be troops able to defend the fortress.
The artillery used in the opening scene is not field artillery, The wheels are too small for the guns to be dragged by horses over ground. The guns are fortress artillery where they are moved only short distances on hard surfaces. The guns would not be set out as seen in the film. There are no limbers or horse teams nearby, nor any infantry to protect the gunners.