Don't leave the old carpet under the new. That will look like an after thought job.
If you are installing new carpet remove as much cabinetry as possible using the reinstalled cabinets as a finish point. If not you will have to use TACK LESS strips around the edges and it will be very hard to get the carpet tight with such short working areas. Leaving the sink cabinets you will be able to remove the beds, table, vanity, and tv stand in about an hour. You could leave the couch and just remove the drawer, but the couch comes out quite easily and the additional working area you will like.
Now having said that I would suggest you consider a manufactured flooring with 3 kids and a dog. Same installation setup but a lot more durable and cleaning is a breeze. See you can have the kids help.
WBCCI 12156 AIR 3144 WACHUNG TAC NJ6
2004 Excursion 4x4
1991 34 ft. Excella +220,000 miles, new laminated flooring, new upholstery, new 3200 lbs axles