First, we say that we have only a finite number of atoms without anything known about them. Then in Appendix A we say that we can have an arbitrary amount of atoms with precisely-equal valuation. This is already suspicious.
Please elaborate. I don't see anything suspicious in your paraphrase. For example, it makes sense to me that, if we don't know anything about the atoms, then we have the same knowledge about all of them, which corresponds to assigning equal valuation to all of them.
It makes sense, but it is no obligation.
The statement of the theorem is "whatever our function does, if it is consistent with the axioms, it is addition". This is used in the context of finite and quite imaginable amount of atoms. We could ascribe all of them equal valuation, but we can have some knowledge why some are more probable than other ones. But the proof requires us to have a lot of atoms and to be able to find as many equally-valued atoms as we need. Proving some inequalities with existing amount of atoms may need more atoms than we ini...
I've recently been getting into all of this wonderful Information Theory stuff and have come across a paper (thanks to John Salvatier) that was written by Kevin H. Knuth:
Foundations of Inference
The paper sets up some intuitive minimal axioms for quantifying power sets and then (seems to) use them to derive Bayesian probability theory, information gain, and Shannon Entropy. The paper also claims to use less assumptions than both Cox and Kolmogorov when choosing axioms. This seems like a significant foundation/unification. I'd like to hear whether others agree and what parts of the paper you think are the significant contributions.
If a 14 page paper is too long for you, I recommend skipping to the conclusion (starting at the bottom of page 12) where there is a nice picture representation of the axioms and a quick summary of what they imply.