Do all firms in all market structures have anything in common?

I think not.

They all consist of buyers and sellers.

In Principals of Microeconomics courses, economists make several general economic assumptions (perhaps incorrectly I might add) about firms and the behavior of firms. For example, we generally assume that:
1) All firms are profit maximizers.
2) All firms make rational choices.
3) Consumers make rational choices.
4) The marginal product of inputs is almost always decreasing.