Not that I have much to add, and this is a very interesting question but maybe you forgot to mention a notion of a Borel algebra which is the sigma algebra generated by a topology.
The Borel Hierarchy might be of interest, also Meagre sets are related to topologies and also mentioned quite a bit.
Oh, and also a collection of sets that is closed under union and intersection is a distributive lattice. Birkhoff's representation theorem gives a one to one correspondence between distributive lattices and partial orders.
Similarly there is Stone's representation theorem which gives a duality between the category of Boolean algebras and the category of Stone spaces, therefore every boolean algebra is isomorphic to a field of sets.
