Is the house finished yet? The videos only show framework

No, the first phase of construction is complete, which was to assemble the structure and all the mechanics that make it fold and unfold.  This was to test that the mechancial design actually works - which it does!  Funds permitting (Kickstarter campaign running until December 20th 2016, see homepage for link), the house will be completed in the first quarter of 2017. The bamboo floors downstairs, seen in the photographs, only need to be attached to the floor frames, but there is glazing (polycarbonate), solid wall panels, insulation, weatherproofing, fittings and finishes to complete - which includes the kitchen, bathroom and rooftop garden.

Where does the furniture and stuff go when the house is folded down?

There is space in the kitchen and bathroom areas to store furniture and belongings when the house is folded down.  By necessity, furniture will need to be minimalist, fold-down and/or multi-pupose.  Some working surfaces could be integrated into the central structure or wall and floor panels, such as fold-out table surfaces.  Additional storage would also be available in the transport vehicle.  Stuff would not be accesible when folded up - the design is best suited to longer stays, rather than moving around continuously as you could with a regular tiny house.

How much does the unit weigh, and how would it be transported?

The unit weighs around 3.5 tons, so it can be towed on a trailer in the US using a pickup truck.  In Europe, the towing limit is 3.5 tons so, while it is possible to reduce the unit weight to around 2.7 tons for towing using aluminium structural members instead of timber, the easiest and cheapest option (for now) is to put it on the back of a 7.5 ton truck and get a category C license (only took me half a day and £350 in the UK).  Otherwise, it could be transported using regular container transport by road, rail or sea.

How much heating would it need in winter?

It really depends how much of the shell is comprised of windows and what winter temperatures you're talking about.  Using 50mm (2") rigid insulation on all floor, wall and roof panels, the heating requirement at 0C (32F) would be about 1kW - but this assumes a completely airtight structure.  Realistically, with window areas minimised or shuttered and the remainder of the shell insulated with 100mm (4") insulation, about 1 to 2kW of heating would be needed at 0C (32F) to maintain a comfortable internal temperature, which could easily be provided by a wood burner.