Everyone is different on this, there is no right or wrong way.
personally, I leave all my fabric in the largest sizes they are in, because I do many modern quilts that can require larger pieces of fabric. If I cut up all my scraps into 2 1/2" strips or squares, what would I do when I wanted to use that fabric in a design that called for 8" squares??
Regarding storing your stash, I first separate mine between solids and prints. Then I sort them by color in each of those 2 groups.
If you think you will make most of your quilts from smaller, standard size strips and squares, go to Bonnie Hunter's Quiltville website and read up on her scrappy cutting/storage method. Lots of folks swear by her plan.
I definitely would not pre-sew HST unless you have a specific pattern in mind.