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When you wanted to put your clothes away the machine would lift all of the layers up until you got to an empty one. Put the shirt down, it snaps a picture of it and catalogs it. When you want to get dressed, you flick through your catalog, select a shirt and it lifts all of the layers to get to that article of clothing. Essentially you wouldn't need to fold clothes, simply lay them out.
You should turn your theoretical project into a company. People will pay a lot of money for innovation and convenience.
In the UK we have washer / dryers. They're not great. They tend to be condensing, not vented, dryers. (Vented dryers are better) And there's more to go wrong. Separate machines are better.
Like you said, it's the folding that's hard.
The model in that article is only 2.44 cf. That is to be expected though, seeing as I don't know of anyone else making a similar device, but we still need to improve before sales would make it worth manufacturing in mass.
I think a more realistic solution to fully automating laundry would be something like the LG TROMM Styler. It's a wardrobe that dry-cleans clothes.
Why is it any more difficult to build a machine that works on a 120V 15A circuit? I never studied circuits, so consider me a layman.
Unfortunately, however, robots might be the most practical solution to this problem.
It could definitely be achieved, it's just a matter at engineering a new type of washer/dryer. What I don't understand is why washers and dryers are separate entities. Yes, engineering a machine that both washes and drys clothes would be difficult, but I'm confident that it could be done.
This is definitely one of the more difficult options!