… what exactly do Rand and Peikoff include in their ontology, and where is it articulated (if it is at all)?
Metaphysically, whatever is is. In terms of being, the axioms (of which that is #3) and the law of causality are almost the whole of Objectivist metaphysics. There is no ontology (as you seem to mean it) in Objectivism. An examination of what sort of things exist is entirely for all other sciences to deal with. Other than recognising that there is such a thing as consciousness (axiom #2), we do not dream of what things may exist in heaven and earth in our philosophy.
Moreover - I am not a professional philosopher so I could be wrong - isn’t ontology more to do with categories of being per se (eg the concept of characteristic, relative location, etc, which gives rise to adjectives and adverbs etc in grammar) rather than some sort of taxonomy of existents as you seem to be asking about? What you are asking about sounds more like cosmology than ontology. Miss Rand rejected cosmology outright as a philosophical subject - see Journals of Ayn Rand, pp698-9.
At least one implication of Peikoff’s statement is that mereological sums exist…
Objectivism recognises that things are the sum of their parts, but also that they exist in a definite pattern that integrates those parts to form a coherent structure. But Objectivism also recognises that while such structures are metaphysically real as the product of what is possible for combinations of fundamental particles acting according to their nature and that this arises from the operation of the laws of identity and causality, our identification of them is a point in epistemology as well as metaphysics. Entities are metaphysically real, but what we physically observe is only one level of comprehension of what exists. That level we call perceptual, where what exists above and below this level must be inferred conceptually. To use the terminology you use at the end of your post, the perceptual level is just that degree of fusion automatically performed by our senses and the basic parts of our consciousnesses (it is in part a learned skill, though learned in babyhood very quickly). The rest of fusion has to be done volitionally and conceptually.
Were there some other creature who was at a different scale of existence, eg some bizarre being hundreds of kilometres across and whose visual and tactile resolution was measured in tens of metres instead of our tenths of millimetres, what we would call perceptual would have to be conceptually inferred by that creature, and what we’d have to conceptually infer on a larger scale than the perceptual would be visually obvious to that creature. Irrespective of what any consciousness’s actual perceptual level is, everything from fundamental particles to the whole of the universe is fully real as existents. Metaphysically, it is all equally real. Picking things out of that gamut for further consideration - for ‘fusion’ - is an epistemological issue. I second Mister Swig’s recommendation to read ItOE.
So we know that Rand permits the existance of composite objects.
“Permits”!?!? This attitude of ‘permitting’ things to exist something she condemned! Again, what is is , whether one wants to ‘permit’ it or not.
Is there anywhere in Rand’s corpus that we can find an explicit account of what sorts of things exist?
There is no such account anywhere, and nor would she provide one were she expressly asked to do so for the reasons I have already said: examination of what exists in particular is for the other sciences. It’s inclusion in philosophy by others was a grave mistake that lead to the dichotomy of rationalism versus empiricism, she wrote.
Does Objectivism permit the possibility of gunk, infinitely divisible entities?
Again with this ‘permitting’ thing…
One may divide something conceptually as far as one likes, so long as there is a metaphysical basis for it. Such division need not be on the entity level, either. Theoretically, one could break up the characteristics of even the fundamental particles for separate consideration, but this is not to imply the existence of something akin to Leibniz’ monads. The only limit is what the nature of the fundamental particles allow - and specification of the detail is, again, not a properly philosophical subject and is instead left to physics.
As to infinity, Objectivism does reject that as a physical possibility, and refers instead to indefinitude. Even the existence of fundamental particles per se is not a philosophical subject. There is only recognition that the infinite cannot physically exist, which just happens to lead one quickly to the conclusion that some such bits-of-something are indeed fundamental. Beyond that, who knows? These bits of something may not even be matter, for all anyone can say.
Does she permit unrestricted fusion of entities, or does she have conditions on what constitutes a parthood relation?
Anything real can be integrated in mind and form a valid concept corresponding to what actually exists. In this instance, what does the permitting is the actual nature of what is being integrated. Fusion by consciousness to form concepts of entities is epistemological, not metaphysical, but must conform to what the metaphysically given lets be fused. Moreover, we may fuse in any which way we like - so long as there is an objective basis for doing so, which basis must also be drawn from the metaphysically given so as to meet our needs for comprehension 
JJM