You draw the cards of all your homes. Does where you live among your properties make a difference, beyond what shows up on your profile?
The property you live in determines the maximum number of cards you can keep in your “hand” at one time. A few special properties marked “Remote Lodgings” in the description prevent you from drawing any card that begins “City Vices:” while you are living in them.
The only Remote Lodgings available are three exclusive to Christmas and one from one ending of The Haunting of the Marsh-House Exceptional Story. Aside from those, there’s no difference in what cards you draw when living in any lodgings. (Though where in London you draw the cards matters of course. For drawing a specific card the Forgotten Quarter or Labyrinth of Tigers is best since they don’t have any location cards; for pure profit both of the Lodgings-specific cards are excellent.)