When I said the cheaper hotels had this facility, I really meant it, particularly those without private toilet and shower facilities.
Of those with ensuite bathrooms (that I actually know about) I'd suggest
Asia Center http://www.asiacenter.or.jp/eng/facilities/
Hotel Villa Fontaine Hamamatsucho http://www.hvf.jp/eng/hamamatsucho.php
Comfort Inn Higashi Nihonbashi http://www.comfortinn.com/hotel-tokyo-japan-JP061

But if you tell us which hotel you are wanting to stay at, we should be able to tell you where the nearest laundromat is before you commit to book there. We may even be able to show you where it is on a map.
The Sheraton Miyako, for instance, has a coin operated laundromat just around the corner. And the Wash & Fold in Yoyogi (perhaps the nicest self serve laundromat in Tokyo) is just a couple of minutes walk from the Park Hyatt in Shinjuku.
Or have a real Tokyo experience and enjoy an urban onsen bath (these are rapidly dwindling away) at the same time as you do your laundry by combining a visit to both.
But sure, you can wash and dry your clothes in the hotel bathroom - assuming you have one - many hotels have a line fixed at one end of the bathroom which you pull out and attach to the wall on the opposite side to facilitate just such an endeavour.
You'll definitely be able to take plastic cutlery on board, you may even be able to take metal utensils - ask on the Travelbuzz forum for up-to-date advice. Better still, ask on the relevant airline forum once you know who you're flying with.
A growing number of Japanese people take their own chopsticks around with them to restaurants as they prefer not to use the ubiquitous disposable chopsticks - don't like them or find them ethically unsound. As RichardInSF says, most places will have metal cutlery, but I still think you should keep all your options open by taking along your own just in case you find that rare place that doesn't (which would likely be a very interesting place to eat!).