This is a great post. +1 for potatoes and onions. (Just don't store them together, they spoil each other faster)

One thing I'll add is that if the OP is in the United States, commercially sold eggs have the bloom washed off them before they get to store shelves, so they MUST be refrigerated.

You can get dehydrated egg products that are shelf stable until you use them. They're not perfect but for most purposes they're good enough.