I believe most open at the same time -- 9 or 9:30 a.m. When my DW and I go we usually get there about 9:30 and the stores are already open and people milling around.
We usually avoid DTD if it's raining -- seems like everyone thinks that's the place to be on a rainy Disney day. So we pick a non-park day to shop, eat, shop more, eat more -- well you get the picture!