Does this actually happen? If a community opened up a co-op shop that started eating into the revenue of a dollar store, would the dollar store company try to fight back, or would they just exit that market?

Yes, I guess well capitalize companies could offer unrealistically low prices, but on the other hand, any kind of co-op or community driven organization has the benefit of not needing the margins. Dollar store investors are there to make a buck, if their capital isn't getting reasonable returns will ultimately exit the business and move somewhere else.