The ISS (as celtschk mentions in the comments) employs a lot of solutions that can give us an idea of what problems have to be solved in order to get furniture to work.
Lets start with the nasties -
A toilet - in SPAAAAACE[2]
- Sucks waste using air flow
- Has gender-specific receptacles for urine and apparently men can use these at a distance, while there's holes on the one intended for women to prevent excess suction
- For poo, you have to strap yourself to the toilet. (at least on a space shuttle)[1][3]
- Currently, the waste is sterilized and kept in bags, which are kept in drawers (with the exception of liquid waste sometimes, which is released into space), but feel free to just get rid of it all - I think they just keep it for studying.
How to not look like me
If you're a clean-shaver or want to shave your legs (I guess), you can use a regular razor[5] but you have to use AstroEdgeⓇ
If you have to trim, you need to use a vacuum cleaner - I think I saw Chris Hadfield use a hybrid shaving machine and vacuum device, pretty neat.
To cut your nails, yep, you use suction again[6] - Hadfield uses a vent (lol) but I guess you can just use the toilet or something.
To brush your teeth, you can use a regular toothbrush and toothpaste but on the ISS, they just dampen it with water, swallow the toothpaste when done and rinse it with their mouth. The problem with a sink would be, the water would get everywhere (remember to brush as long as you can sing happy birthday![7]) I assume that you could have a toothbrush that secretes the water and paste and some specialized means of cleaning it out, but you'd have to use a toilet funnel to spit out the water safely.
Sometimes you just need a shower
For hair: On the ISS they use regular water and no-rinse shampoo. Apparently some use towels to scrub a bit. To prevent humidity, you need air conditioning :P. I assume that you could improve things a bit by packaging the shampoo and water together, but overall, no showers as we know them.[8]
For the rest: Sponge baths with moist sponges and get it out with a wet towel.[9]
The problem remains - water gets everywhere and the excess humidity would be a huge problem, so you make the process as efficient as possible. If you don't however have an issue with water supply, power to pump the water to high pressure and your air conditioning is super-efficient, you could probably have a regular shower.
You can't be up all day
Sleeping without gravity means, as usual, strapping yourself in. Sleeping bags are efficient for this since you can confine yourself and stay warm.[9]
Eating means again, strapping yourself to the chair.[9]
To drink, you use a bag with a straw.
Since you're not feeling the weight of your body, you don't really need a couch to sit on when watching TV, so you just float in place.
Astronauts have to use special food bags, prepared for rehydration (like Marty's mom in BTTF2, although noone can rehydrate a pizza like her!). Stick in the wall and pump it with water.[11]
How does all this come together if you're not starved for space and supplies?
Rooms: Not very hard - regular sliding doors would work just the same and probably would be preferrable. Regular doors don't depend on gravity besides, in some cases, to define whether they're going to open more or close if they're left loose. I'd imagine that if you're in space, or in a general zero-G environment, using the regular air pump systems you see in a lot of places to make sure a door closes would work fine. If you want a door that can stay open instead of float back and forth, you could use a stopper, much like how large gates have: there's an arc rail on the floor with holes and you open the door however much you want, then sink the stopper into the hole.
Bedding: You're probably limited to the astronaut solution. The problem is keeping yourself from floating around while sleeping, so if you fall asleep on your desk you might float out of the room or something. A simple strap would prevent you from floating too far, so you just get into the habit of using your new, fancy Nike desk strap. This means sleeping bags attached to surfaces, no matter the orientation, are most preferable and convenient. They can be made large for comfort and have a velcro on them for your water bag, in case you need a drink at night.
Eating: Again, just use your fancy Nike strap (might be a good idea to wear these and just hook them wherever you need instead of having one for each piece of furniture). Sit down, strap in, eat up. Liquids go in bags, solid stuff is packed together. I imagine sandwiches will be very popular. Spoons aren't obsolete, since peanut butter is still around and 2nd-3rd generation children growing up in space will probably be able to eat soup using a spoon - assuming the soup stays in a plate :P.
Since you don't want to eat rehydrated pizza forever, you'd have a kitchen. Instead of cooking in a pot, you'd probably have a kitchen on the wall, with modules equivalent to pots and pans, stirring included! Glass allows you to see what's going on but everything stays sealed in until it's ready. Put your veggies in a slot that flows through an airlock into your square pot, inject water through the water supply, inject salt through the appropriate button that links to your salt module and give it a temperature. Once it's done, tell it to suck it into a bag or dry it out and send it to your oven module, where you've already prepared a pan. Let it cool down and then take out your pie - use a fork. I assume kitchen appliances would be much the same overall, but with a bit more automation to prevent spillage and make things easier to control. The market would be full of tiny modules for this and that. You can get a full cooking process, just like home, but on the wall.
Recreation: Again, just a matter of partitioning space. Couches can be done away with (use your Nike strap) and everything can be velcroed in place if necessary. Stuff that doesn't have to be moved can be bolted or screwed in place.
Storage: The real question is, how do you keep your stuff in place? Cupboards are easy - however again, the astronaut solution is probably best. Get everything velcroed - possibly storage will have velcro on all interior surfaces (when intended for regular people stuff) and items come with velcro attached or you have bags of various sizes to put your stuff in and the bag carries the velcro - little difference from drawers and drawer sizes and shapes. Velcro isn't magnetic, so no problems there - it works for everything.
Toilets: As seen above, if you can handle the excess humidity and have enough water and enough power to pump it, you can just have a regular shower. Even water spillage wouldn't be as much of a problem. Still, you'd probably have hybrid cleaning tools to make things easier and faster. A bubble bath might still be out of the question but you get a zero-G shower, which can be just as fun - the point of a bubble bath is partially the relaxation, I assume a lack of gravity can have a similar effect.
For waste, the existing solutions work well enough and I don't think much more than appearance would be different. You might have a luxury bathroom that resembles an Earth one, but it would work much the same. Air vacuums, bags for storage or shoot the waste into space and vents to prevent your hair and nail clippings from choking everyone and floating into electronics.
