Modern conveniences are a nice thing. Ask anyone that grew up in the 40's or 50's and they will tell you that they don't miss hand washing clothing. My mother used to tell me how she would run her hands through the clothes wringer and how it hurt badly. A lot of what we have today even in a huge 750 sq foot home is something that our grand parents would have wanted when they were our age.
Living in a tiny home though means giving that up, a Laundry in your home is a big luxury you can not have. Well a traditional one at least. Unless you build your home larger to make room for it. A stacking washer and dryer, apartment small size is 30"X30" by 72" tall and it needs 6" at the back and 3 inches on each side. Plus you need either 220Vac power or a large propane tank/natural gas hook up. This means you are not in a trailer but on a permanent lot. Even the low power ones draw so much power that the solar system needed would be expensive as all get out.
So what can you do? look for alternatives. Laundrymat is always an option, and give you and excuse to be gone for a few hours once a week. If it's summer, you can buy a wonderwash hand crank washer and then air dry on clotheslines, I hope it's windy and dry or it will take a couple of days to dry. Or you can do like your grandma did, get a washboard, buckets, wringer, and do it all by hand. I have not tried the wonderwash, but many reviews online say it works great but you still need a wringer of some type. In the winter you can hang clothes to dry in the home, people did it for centuries.
Dishwasher, honestly you can get a tiny counter top dishwasher, but in a micro home you don't have enough dishes to need a dish washer. If two of you live there, you have 3 coffee cups. His, hers, and a guest cup. you just don't have the space to keep service for 12. hand wash your plate when you are done eating makes a big difference. Also smaller dinnerware is a bonus. Eating off of 9" diameter plates makes your dinner look huge compared to what is normal today.
Refrigeration is another one. if you can generate the 100VAC for a dorm fridge, you have it solved for cheap. Otherwise your choice is a RV two or three way power fridge that runs on 12V, 110AC, or propane. Problem is these need to be on an outside wall with ducting for exhaust. and they are very expensive compared to a large dorm fridge.
Entertainment is easy. iPads can cover most of what you need, but if you are a gamer geek, then new LED Tv's are incredibly thin and a Xbox is a multi use device. a 32" TV in a micro home where you are sitting 4 feet away looks huge. and you can use it for your laptop screen as well. The Xbox acts as a game system, DVD player, and if you have internet, netflix, hulu, etc... Hey when it's raining outside or you have 30 feet of snow on the ground, You will want to do something like playing a game or watching a movie. A better choice though is a laptop and many people will use that for everything including gaming and movies. They dont want the 32" TV and xbox. and that saves you money and space. a nice 17" laptop 18 inches away from your face feels like a 98" TV screen. and if you want surround sound, they sell 7.1 surround headphones so you can get the theater experience.. you can even get 3D display laptops to watch 3d movies on.