What axioms (other than choice) have a taming effect on the ordering of cardinalities? Axiom of choice arranges all cardinalities into a well-ordered chain but without it their ordering can be wild in general ZF models, e.g. two cardinalities may not even have inf or sup. However, according to Woodin in $ZF+DC+AD_{\mathbb{R}}$ low uncountable cardinalities form a nice lattice, and according to Caicedo and Ketchersid in $ZF+AD^++V=L\big(\mathcal{P}(\mathbb{R})\big)$ if $S$ is strictly larger than a well-ordered cardinal $\kappa$, then either $\kappa^+$ or $\kappa\cup\mathbb{R}$ embeds into $S$, and "$\omega_1$ and $\mathbb{R}$ form a basis for the uncountable cardinals" (not sure what basis means in this context). 
What other axioms/ZF models are known to have some taming effect on the ordering of cardinalities? Do weaker forms of choice like "every set is linearly orderable" have such effect? References appreciated.
 A: I don't have anything in mind which "tames" the cardinals structure. I also think that you don't fully understand the consequences of $\sf AD$ on the cardinal structure if you say that it tames the structure down. (Note that even Woodin, in the abstract you reference, points out that the cardinal structure is very complicated.)
But let me just point out the following. The axiom "Every set is linearly orderable" follows from the Boolean Prime Ideal theorem, which is shown to hold in Cohen's first model. In that model there exists a Dedekind-finite set. Therefore the example I gave you in the math.SE question about the failure of infimum of two sets holds in that model.
Now, without sitting to verify the details, here's a suggestion as to how to destroy the supremum property while still having "Every set is linearly orderable". By adding $\aleph_0$ Cohen reals, and $\aleph_1$ Cohen subsets of $\omega_1$, and taking automorphisms of the forcing which look like the ones used in Cohen's first model (namely, an automorphism of a disjoint union of $\omega$ and $\omega_1$ which doesn't "mix" the two parts), and similar ideal and so on, one should be able to get a model in which there are two incomparable Dedekind-finite sets, but every set can be linearly ordered. In that case, the generalization of the example in which $\sup$ fails can be executed.
(I should say that I don't remember at the moment whether or not the old question "If there is an infinite Dedekind-finite cardinal, then there are two incomparable Dedekind-finite cardinals" is answered. In case that the answer is positive, then Cohen's first model is a model for the failure of $\sup$ for cardinals as well.)
