Do birds ever fly around inside the store.

Birds, bats, flies, mosquitoes, even the occasional pteradactyl. But thats pretty rare.

Sometimes

It doesn't happen very often.

No. Once they get inside, they stop by and pick up a cart, just like everyone else.

Any store with automatic doors can have this happen.

Yes. That can happen with any large store with high ceilings. It's not common though.