In Luke 10:6, the apostles were told to stay at the house of a "man of peace." What if the house was not occupied by a man of peace? Might not that man still offer hospitality? What does "peace" really mean?

A man of peace is a semitic expression meaning "a peaceful man." In this context, a welcoming man, someone who is open to your sharing your message. You say shalom to him and he reciprocates. I doubt very much that someone inimical to the apostles would ever offer them hospitality. After all, his house was to become a beachhead for the proclamation of a new unfamiliar, challenging, and (sooner or later) socially unpopular message.