Does Leeds Castle have rooms set up with furniture as it was back in the day or is it more barren like Hampton Court Palace?

Yes it does.

The Queen's Gallery, Bedroom and Bath (Catherine, Henry V's queen) and the Banquet Hall, from Henry VIII are beautifully furnished from the period. Leeds is very much worth a visit!

And I believe the owners live on site in a adjacent building.

And it has a famous dog-collar museum!

Leeds Castle was restored in the early 20th century and many of the rooms are now furnished in the 1940's decor. There are, however, a few rooms in the older part of the castle furnished in the period of Henry VIII. It is a very nice (and easy) day trip from London.

Besides the castle (beautiful setting), there is a duckery, gardens, an aviary and a very difficult hedge maze.

If you will be there in the summer, they have concerts on the lawn. We attended an exellent concert there last June. The weather didn't cooperate, but it was a wonderful concert.

At Leeds Castle you will also find the most beautiful floral arrangements in some of the rooms.
The owner was an American who died a couple of years ago and left instructions in her Will that these arrangements be done each week using flowers from the absolutely wonderful gardens.