I wash before cutting EVERYTHING that is washable (except the battings that specifically say DO NOT prewash or the polyester battings like Fairfield? that are not needlepunched)
Most of the fabrics come out of the dryer smooth enough to "hand-press" while folding them. I press/iron right before cutting - that way I get rid of the fold lines that I feel interfere with cutting accuracy.
My stash is "organized" - for lack of a better word - by the following - large pieces (usually three yards or more) are folded on a bed - I Spy/novelty fabrics are in plastic totes - Cat Prints are in totes - Black and White fabrics are in plastic totes - the "colors" are more or less together on shelves. Small pieces of various colors are sorted into small totes.
And then there are the couple of totes of - almost trash scrap. The pieces are big enough to make string blocks or crumb blocks - but mostly annoying.
Take into consideration that I've been accumulating for over 20 years.
My advice NOW is to spend sensibly on fabric. I went totally crazy for a while. (I've cut waaaaay back for the last 8 years - but it's still hard for me to pass up a good clearance sale)