The eigenvalues of elements of infinite order are certainly not trivial to study,
and as far as I know little has been determined about them. Keep in mind that arbitrary infinite Coxeter groups are quite varied and hard to study systematically beyond the most basic theory.

Maybe I can clarify at least what you've summarized about the special case of Coxeter elements. These were originally studied by Coxeter for finite reflection groups, where the eigenvalues have remarkable properties. But the definition makes sense for any Coxeter group having a finite set $S$ of involutions as standard generators. You just take the product of elements of $S$ is any order to get a "Coxeter element". The problem is that these are not usually all conjugate, unless the Coxeter graph is a forest (union of trees, where there are no cycles).

Most of the affine Weyl groups (irreducible by definition) have trees as graphs, but then it's still tricky to study the eigenvalues of a Coxeter element. The only significant work I'm aware of in cases where the Coxeter elements are all conjugate was done by N. A'Campo: Sur les valeurs propres de la transformation de Coxeter. *Invent. Math.* 33 (1976), no. 1, 61–67. (This should be available online at http://gdz.sub.uni-goettingen.de/gdz/.)

The main result A'Campo obtains in this note for Coxeter elements outside the affine Weyl group case is that all real parts of square roots of eigenvalues are $>1$.

Probably you haven't missed any important literature (though I'm not certain). The question you raise is definitely nontrivial in any case and might be best approached in narrowly defined cases.