In my experience, category theory is very successful at providing powerful machinery to reason about large objects or objects unrestricted in size, for example (logical) models (via accessible categories) or "nice" topological spaces (via simplicial sets). However I haven't seen many impressive applications of category theory to categories of finite objects, for example finite graphs or more generally finite models, whose behaviors can be radically different from the corresponding objects of unrestricted size. I certainly haven't read widely enough, but I want to ask anyway: is this finite/infinite divide a conception or misconception? If the former, are there any intuitive explanations? If the latter, what are some convincing counterexamples (for example, applications that provided major insight in what graph theorists would be interested in)?
I would say you could make good headway on this by looking over some of the research projects of Tom Leinster, Mark Meckes, and Simon Willerton (and others I may be forgetting), centering on various notions of size/magnitude for graphs, finite metric spaces, and various other structures. You could start by perusing the entries under "Size" on Leinster's webpage. Many of the entries with link to "discussions" refer to discussions taking place at the $n$-Category Café over a period of years.
I don't exactly understand the question, but I think you should familiarise yourself with the notion of a fully dualisable object in a (possibly higher) symmetric monoidal category. For example, a vector space is fully dualisable iff it is finite dimensional. In the bicategory of algebras and bimodules, the separable algebras are fully dualisable. This notion often captures desirable finiteness conditions.