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.

Cooperatives do not get rid of the net negative cycle. Ultimately whatever the benevolent entity ends up being, it becomes a contest of who can bear to lose more money.

Cooperatives distribute the losses but it is still a money pit.