As I understand it, although the Duke of Lancaster is the Monarch … it’s a completely separate inheritance and administered by the Duchy Council and not the Crown Estates. I think the holder ’only’ gets the net income for the year (which I think The Queen started to pay tax on in the late 80’s/early 90’s even though it’s technically exempt). If the Queen abdicated, them the profits would go to the new Monarch, in the same way the Cornwall one would pass down.
if the Monarchy were to be abolished then technically it should pass to the next in line for succession of the Duke of Lancaster regardless - although our old friend Oliver Cromwell took possession, leaving Charles II temporarily Duchy-less, and flogged a lot of it off during his Protectorship.
Staff are the only ones to live in on a permanent basis, I think.