The best resource I've seen is
http://oasisdesign.net/greywater/ and
Art Ludwig's book
Create an Oasis with Grey Water.
From the book and the many informative things on the website, I understood two things in reference to your questions.
1) Filtering greywater or trying to clean it for a second use is very difficult, and very likely to clog up and require many nasty sessions of clearing work again and again.
2) Storing greywater makes it so stinky that it may actually be considered blackwater. In my
experience, greywater stored for a day does indeed stink. I only use it for irrigation at ground level so I don't care, but yeah, it stinks if stored.
The simplest way to use greywater is to direct it as quickly as possible to the topsoil, and a very effective way to do that is mulch basins or mulch filled channels.
Here in Ladakh, people used to wash dishes with warm water but no
soap, and give the used dishwater and food scraps to the animals.