Interesting question. Furniture literally means things provided (for comfortable living) so from that point of view all of those things, except perhaps paintings which are decorations or ornaments, are "furniture". But I would say most people think of furniture as being chairs, tables, chests, desks etc. - movable items traditionally made of wood (and of course in most other European languages "movables" are what they are called). If kitchen appliances had existed when "furniture" was first coined, who knows if they would have been included? I think perhaps they might have been.
I think in usage though, carpets, curtains, kitchen appliances, and certainly pictures, are not thought of being furniture as such.
According to my conception of "furniture", it includes none of these. If someone told me that an apartment was unfurnished, say, I would know that there are no tables, chairs, lamps, or beds, for example. But I may then ask if there is a refrigerator, air conditioner, washer, dryer, etc. I would assume that there would be no curtain, carpet, or pictures on the wall because these typically do not exist in unfurnished apartments, although they are not pieces of furniture.
I think when "furnished" is used as an adjective it has a wider meaning than "furniture" does. If I was told a flat was furnished, I would expect carpets, curtains, kitchen appliances, light shades, etc. (and even, perhaps, pictures).
However, if you were to list the contents of a house, I think you would not just say "furniture", but would have to say "furniture, kitchen appliances, carpets, curtains, [etc.]"
When I purchased my "unfurnished" home, it included appliances (refrigerator, clothes washer, clothes drier, dish washer), but it did not include any furniture or decorations (pictures). Rugs would not be included; wall to wall carpet would be included.
Packard makes very good points. A furniture store sells furniture: chairs, sofas, beds. They have sub-genres of furniture, like accents: coffee tables, end tables. They have home accessories scattered throughout their stores: paintings, lamps, mirrors. All these qualify as furniture, I think.
An appliance store sells stoves, refrigerators, dishwashers, clothes washers/dryers.
A carpet store sells carpet, although I've been in some furniture stores that have a Flooring Section.
My opinion would be that all those items listed in casino's post are not furniture.