I greatly apologize in advance if this is not a well formed question, and feel free to redirect this in any way.
So I have begun reading Logic as Algebra by Halmos and Givant. In their book, they provide an algebraic development of mathematical logic. I haven't gotten very far, but I do see how they change the concepts of connectives to binary operations, inference rules as a function, etc. This brings me to my first question. Can algebra be reinstated as some foundations of mathematics, much like set theory? Could I very well just assume some (albeit nowhere near as elegant) axioms equivalent to ZFC in terms of algebra and develop mathematics from there? Furthermore, I would very much like to continue reading on algebraic logic, but from my own search, the literature isn't entirely as dense as it is in other subjects. One of the few that I have seen is An Algebraic Introduction to Mathematical Logic by Barnes. Is this a good book?
My last (but not least) question is based off of the spirit of the first two. I am gaining interest into graph theory, and I was wondering if there was any analogue of a graph-theoretic development of logic as with the algebraic one I mentioned above. If so, (and assuming the answer to the algebraic foundation question above is yes) could you also formulate an axiomatic development of mathematics using graph theory? These last few questions on graph theory are based on a potential project I would like to work on, so I would appreciate any help in the matter.