Question: It seems like a glaring factual error that Balian is able to read and write, given that he was raised as a peasant in the Middle Ages. Did I miss some mention as to how he became literate?
Answer: As Sereenie said in the corrections, Balian has spent months travelling with his father, during which time he has learned many things, including sword-fighting, and how to read and write.
Question: In the scene where Balian arrives at Jerusalem, at the peak where Jesus was crucified, as he sits there, is there a strange shadow of a man crossing in the background or did I see wrong?
Answer: When Balian thinks to himself, "God, what is it you want of me?" in the next shot it is not a "strange shadow" of just one man, there are a few people who walk from left to right or right to left. The cinematographic effects of this specific shot gives an ethereal quality to the movements of these people.
Question: How did Balian, a peasant, get to be literate at the beginning of the film, before he leaves with his father? He reads what's carved into the wooden beam over his workplace.
Answer: Who says he was reading it, someone could have told him what it said, so he knew what the beam says rather than read it.