This was both a greeting and a blessing. Here "house" refers to those who live in the house. AT: "May the people in this household receive peace" (See: [[rc://en/ta/man/translate/figs-metonymy]])
Here "peace" is described as a living thing that can choose where to stay. AT: "he will have the peace you blessed him with" (See: [[rc://en/ta/man/translate/figs-personification]])
It may be helpful to restate the entire phrase. AT: "if there is no person of peace there" or "if the owner of the house is not a peaceful person" (See: [[rc://en/ta/man/translate/figs-ellipsis]])
Here "peace" is described as a living thing that can choose to leave. AT: "you will have that peace" or "he will not receive the peace you blessed him with" (See: [[rc://en/ta/man/translate/figs-personification]])
Jesus was not saying that they should stay in the house all day, but that they should sleep at the same house every night they were there. AT: "Continue to sleep at that house"
This is a general principle that Jesus was applying to the men he was sending out. Since they would be teaching and healing the people, the people should provide them with a place to stay and with food.
Moving around from house to house means going to different houses. It can be made clear that he was talking about staying overnight at different houses. "Do not go sleep at a different house each night" (See: [[rc://en/ta/man/translate/figs-idiom]] and [[rc://en/ta/man/translate/figs-explicit]])