How can one respond to the misconception that Muslims worship the Black Stone? And can the Prophet, peace be upon him, err, despite being infallible?

The prophets are infallible concerning grave sins and in conveying God's message. Minor sins may occur from them, but they are not condoned for them. Rather, God alerts them to these sins, and they repent from them. One must be cautious about taking a non-Muslim as a friend.