I have to admit that jumping between the various parts of the question as well as the editing/comments almost confused me, or at least gave me some discomfort. So I try to clarify the situation although I am fond of reading some of the previous answers, in particular that given by a_cyclohexane_molecule. To this end, I go through your text. Please note that capital is not shouting but an attention driver.
1 - "But strong bases dissociate completely in solution, so I would guess this means that such hydroxides dissolve only a small amount, but the dissolved portion dissociates completely."
Yes, this is the meaning and what happens.
2 - ".... problem with this interpretation is that these hydroxides are ionic compounds, and my understanding is that "dissolve" is essentially equivalent to "dissociate" for ionic compounds, i.e. ionic compounds dissolve by dissociating.
Not completely right. The ACTUAL amount of THESE hydroxides - NaOH and Ca(OH)2 - that goes in solution (solubility) is fully dissociated. It is again point 1), that you inadvertently generalise to all hydroxides, whatever their base strength is.
Think of Me(OH)n in water. Me ions coexist with Me(OH)n-1, Me(OH)n-2 etc. positively charged ions (according to the stoichiometry). In which sense the dissolved hydroxide is fully dissociated? It is not, even if you postulate that what goes in solution undergoes at least one complete first dissociation process, i.e. no neutral entities can dive in solution (I think this is matter for a separated question. We are in the case "to which extent a bond is ionic?", at the end. Or even as to ask what is the biggest neutral agglomerate of ionic compounds that can be filtered out of a solution. It is not a coincidence that most hydroxides are tricky to handle :) Unfortunately real things rarely fit in absolute definition).
3 - " if there's no distinction between dissolving and dissociating for ionic compounds, what does the description "strong base that is only slightly soluble" mean? How does it differ from the term "weak base?"
Here is where some definitively confusion comes in. First question in the paragraph. We answered already. There is a distinction, as above. 100% dissociation does NOT imply that the same amount is in solution. Microscopically, let us take for granted that each whatever small portion of the compound that gets dissolved does dissociate. It does ONLY IF gets dissolved, AND IF is not like the case Me(OH)n above.
Second question in the paragraph. Let me stay in the realm of Arrhenius: a strong base definitively gives you a solution with a high pH.
Differently, a very weak base does not generate a considerable amount of OH-, their final concentration it is about the natural one in water (from Kw). (You get see this in B-L or L theories, too, and you need those theories to rationalise the behaviour of ammonia, for instance).
3 - " If there is a distinction, and it is possible for ionic compounds to dissolve without dissociating, how does such dissolution work? I have a model for the distinction between dissolution and dissociation of covalent compounds: first the compound dissolves into separate molecules, then each molecule dissociates into two or more ions. If ionic compounds can dissolve without dissociating, into what sort of particles do they dissolve? Formula units? Small clusters of ions? Something else?"
I think that answer 2 applies here, too.
The point is that if you stick to hydroxides (except alkaline and earth-alkaline), metal and amphoteric hydroxides are weak bases because
are so sparingly soluble in water that by a practical, Arrhenius based viewpoint, do not gives considerable amount of OH-. This will be true even assuming their complete dissociation!
a finer analysis of their equilibria in water does reveal the coexistence of different degree of dissociation, so they can be classified as weak indipendently of their limited or negligible solubility.
Although the OP did not have troubles in cases in which he could easily identify the covalent molecule that goes in solution without (or better, partially) dissociate, let me conclude by pointing to the fact that a weak base it is not necessarily sparingly soluble (as for a weak acid it is not necessary sparingly soluble), as this will not leave room for ammonia (acetic acid) etc.
Ergo we still need a definition of weak bases independently of their solubility.