We inherited a Swiffer, and I thought it would be worth trying. It wasn't. When you have to scrub a floor, you need something with some substance. A handle that's hollow and smaller 'round than my finger with plastic joints won't cut it.
I happened to have an old metal sheetrock sander gizmo, the kind you can screw a handle into to sand the ceiling when you put in new sheetrock. And, I have a hardwood handle that fits it perfectly. Plus, I have a lot of cloth rags I can reuse to my heart's content. Presto change-o, homemade and better than a Swiffer.
Laundry requires a water container, soap, water, a source of heat, a way to squeeze water out of washed items, and a way to hang up and dry the material being washed.
I bought the makings for homemade laundry soap and a large plastic tub to wash clothes in. I took a look at a plunger washing gizmo and decided that I needed something that wouldn't rust, so I got a clean toilet plunger and a long wood handle to screw into it. I've seen a version of this using a five gallon bucket with a lid and some holes drilled through the plunger, both of which sound like good modifications.
Soap's really mostly a surfactant and dirt attractor to help water do its thing, so you don't need much, but you do need to agitate what you use through the clothes or whatever you are washing. After washing, there's the matter of getting the soap and water out. The soap's easy; add some vinegar to your rinse water. The water's a bit tougher; wringing wet fabric out is hard on the hands and time consuming. So, I found a workaround: someone gave me the wringer from a commercial wringer/bucket combination, and it will fit on a sawhorse or something similar and be used to wring out clothes. Hot water is probably a necessity for things like underwear and diapers, but most things wash well enough in cold water, so heating water for washing won't be as big a production as it might seem.
I Freecycled an umbrella clothesline, and have enough wooden clothespins (nobody else wanted them when we cleaned out a relative's house) to be able to hang up the equivalent of a washer load or two, so the drying part's not a problem. I found a wood collapsible drying rack to use indoors in the winter, but it won't hold much. Washing in winter will be...interesting.
Without running water, we'd have to resort to using a basin and pitcher scheme for hand washing, but fortunately, I remember using that method when we'd use the cabin my folks used to own before it had running water (oddly enough, it was on a lake, but had no running water. Go figure).
Washing dishes is another story altogether. In a short-term emergency, I think that paper plates, as unenvironmental as they are, would be acceptable. In a longer-term emergency, they won't do; you end up using too many of them and all they're good for after use is burning. One-use-only items are not practical PSHTF, with some few exceptions, so it would be back to the dishes. PSHTF, it's the Corian that will get used, not paper plates or fine china.