Goofs

It's never explained how Lukashka and his men return home from the Sultan's palace after the road they took to get to the palace is destroyed.

The road wasn't destroyed but blocked. Though it would take time, the Cossacks would have been able to clear it. Also, there often are different routes between two places, especially on horseback. There is no requirement that a film explain every detail.

It's never explained why Lukashka is apparently the only man in the village who can read and write.

However, it was established that Lukashka spent his early years not doing the things other boys/young men didn't, so while they weren't learning to read and write, he might have been.

It's never explained why the villagers and the Turks are enemies. Ironically, the Cossacks fought for the Turks in the Crimean War, which ended 7 years before Lev Tolstoy published the novel the film is based on.

It's basically stated that the two "tribes" have always fought. This movie is not based on historical fact, so the Crimean War, or any other actual history, is irrelevant - Tolstoy (and the screenwriters and the directors) used "artistic license".

It's inconceivable that the Sultan would mistake Lukashka's letter as coming from the Tsar, and break the peace struck between them by having his men attack the Prince's carriage.