In this case, the private house probably belonged to a distant cousin of Joseph. They were not the only family of the house of David to be traveling to Bethlehem at that time, and families tended to congregate together.
When there is not even any space on the floor you can put the cousins in a tent in the back yard.
Yes, or perhaps even Joseph's immediate family. While Scripture seems clear that Mary was from Nazareth, the notion that Joseph wasn't himself from Bethlehem would appear to be more of an inference than anything else. Luke writes: "And all went to be taxed, every one into his own city." Luke 2:3 (KJV). While it has been supposed that Bethlehem was Joseph's "ancestral" home, as it were, a straightforward reading of the text suggests, alternatively, that Bethlehem was in fact his home town.
Note, too, in this regard, that Matthew's Nativity narrative has Mary and Jesus living in a house, in Bethlehem, at the time of the adoration of the magi, which likely occurred many, many months after Jesus's birth. Matthew 2:11. In other words, by that time, Joseph and Mary had wed and were permanently residing in Bethlehem.
Thereafter, after the flight into Egypt, Joseph's original intention was to return to Judea, but "being warned of God in a dream, he turned aside into parts of Galilee." Matthew 2:22 (KJV).