I wash almost all fabric when it comes in the house. Some of mine comes from dubious sources, such as friends sewing rooms. One piece was covered in cat fur, folded to the inside. I discovered that when my displeased cat started peeing on it.
I clip the corner off the fabric before washing, it reduces fraying and tells me if it's been treated. Exceptions to washing are precuts and the rare occasion when I purchase all the fabrics for a quilt from a single line. They'll all shrink at the same rate. I can seldom afford to buy that way.
I usually press to one side, the Dark Side, open when warranted. Garments I typically construct on the overlocker so pressed to one side.
Huge fan of Best Press! I always press fabric before cutting it, usually with a spritz of BP. Ladies use it when you garment sew. It makes all fabric behave a little bit better.