I just returned from a trip to the Holy Land. Our Christian Jewish native Bethlehemian-guide believes that yes, it was Joseph's extended family's home, crowded with relatives who had returned to Bethlehem to pay their taxes. The lower floor (a cave, and probably the older and original part of the home) was used like our garages: storage, workspace, and to pen the animals. The upper floor was added when possible, and was sleeping and living space. Mary and Joseph would have been given the quite serviceable lower floor out of respect of her need for privacy.
Very interesting: He said that Jewish babies at this time were NOT wrapped in swaddling "clothes." They were dressed in miniature period clothing. Luke 2:7 says "cloths", not "clothes." I've misquoted and mis-taught this for my whole adult life.
Olive oil was pressed, grain was crushed, fabric was woven: daytime work was down in this cooler cave area. What was also made and stored in the lower-floor working area was swaddling cloth, which had 2 purposes.
When a perfect lamb was chosen for sacrifice, it was carefully cleaned, and then it's feet were 'swaddled' together with these specially prepared clean strips of cloth, so that the lamb would remain clean, and it's feet would never touch the ground again. It would be respectfully carried by the head of the family to the synagogue for sacrifice, it's feet bound against any possible of injury to this perfect sacrifice.
The 2nd purpose of swaddling cloth was the wrapping of a body in preparation for burial. In the Jewish culture at that time, burial was within the day (or next day?) Thus, it was necessary that each family have ample stores of clean swaddling cloth prepared and stored for these two purposes.
It's easy to assume that perhaps Mary arrived unprepared for a new baby, and found this clean cloth and used it to wrap her newborn child. And what prophecy! According to the gospels he was to become a perfect sacrifice, and to be wrapped in fresh linens at His death, but leave them neatly folded at His resurrection!