When you say such terms as "base state" and "superposition", are you assuming a preferred basis?

Well, yes and no. The paper itself doesn't assume a preferred basis, but surely to have a completely coherent picture of MWI you need to solve the preferred basis problem.
Decoherence supposedly does that with the concept of pointer basis.

Isn't it the squared norm?

Yes, absolutely! Edited, thanks.

This seems a bit too brief to me. If we wish merely to assert that there is a number that corresponds to the probability of finding oneself in a branch, that is much too trivial an assertion to demand explanation.

Well, the probability of a branch is not just a number given from the theory, is a number derived from a complex amplitude, and this has a number of consequences that makes it non-trivial. Plus, the Born rule in the context of mixed states has a more complex formulation.
In general though, since it's usually (that is, outside of MWI) assumed as an axiom, I guess you could say that it's trivial.

Can you explain the math on this?

I'm afraid not better than the original authors, so if you really want to get the math you'll have to look at the article.