Ilyn Payne cannot read, yet he is a knight. Was illiteracy a common problem among the nobility?

Usually pretty rare. At least some functional literacy was normally considered necessary.

I’m guessing the Paynes are pretty recent nobility – we don’t hear of Paynes before Ser Ilyn, their sigil suggests they used to be tax collectors or accountants for the Lannisters, so maybe Ser Ilyn never learned growing up and got a battlefield promotion to knighthood. It’s quite possible that Ser Ilyn has some kind of learning disability.

It’s also quite possible that Ser Ilyn’s trauma from the tongue removal has meant that he has no desire to communicate. Given his drinking, his low level of self-care, his obsession with his work, I wouldn’t be surprised.