Up to the new owners. The sale includes all the stuff.
Usually? Usually? That’s not answerable, not really. It also varies from place to place and general situation to general situation, and new owner to new owner (and the people they get to help them).
Some people look for treasure. Fewer find treasure.
Many people throw it out or have others throw it out. If someone finds something desirable, they keep it and/or present it to the owner to decide what to do with it.
And yes, sometimes properties are re-sold or rented without clearing out stuff from the previous inhabitants. I know someone who has rented several offices and sold off the stuff from previous tenants. In his case, typically the renters are happy as long as the property isn’t damaged, and have no interest in doing the work of sorting through the stuff, as long as it’s not a legal or financial risk to them (like if it’s dangerous or so awful that it will make their property harder to rent for as much money). For him, or I suppose anyone looking at the kinds of offices in the places he’s looked, that’s very common.
Yes, furniture and books and all sorts of other stuff included.
In case of death with no designated heir, next of kin has the right to it, but other local laws may apply. In general that will have been thought to have been figured out before someone is selling a property.
What’s usual or likely to happen, in my experience varies by location and type of property. Some real estate and rental markets have typical buyers/renters who are more particular about clean dwellings than others.
In Seattle, mostly people expect a house or apartment to be thoroughly cleaned and repainted and have probably little or no stuff in it unless it’s a furnished apartment or the stuff is major good furniture.
Houses often come with at least some “stuff”, such as spare parts and construction materials and paint, old nails, jars, gardening bits and pieces, bottles, and other stuff that “might be useful” or was just forgotten in basements, crawlspaces, outbuildings, yards, attics, closets and other nooks and crannies. It’s all part of the purchase.
Sometimes though a house may have outbuildings which are in such bad condition that they are either likely to be destroyed, or they have a bunch of crap in them that probably ought to be hauled off, and the seller doesn’t choose to do that themselves. There may even be abandoned vehicles or other ruins.
In Chicago, I’ve seen some pretty poorly kept places for rent.
In Britain and Europe, expectations of cleanliness and repair can be MUCH lower, and just factored into the asking price. There I’ve seen repairs needed, holes to outside air, large amounts of dead spiders, lots of old stuff of who knows who etc etc.