Can I store dry ingredients (flour, baking powder, baking soda, etc) together?
As long as they're all dry ingredients, then you should be just fine — after all, that's exactly what a box of packaged cake mix is.

Again, if it's just dry ingredients, I see no need to refrigerate it. I would put it in an airtight container — preferably a glass jar*. Placed in your pantry, it should have a shelf-life of a least a couple of months.

&#42; Glass jars are my preference for storing dry ingredients — they're easy sanitize, and the twist-off cover makes an airtight seal: