The first question is the main topic of Reggio's recent paper Polyadic Sets and Homomorphism Counting, which is also a good reference for previous results in the literature (including that of Pultr mentioned in the comments). In particular, see Theorems 4.3, 5.10, and 5.12, the latter of which generalise beyond locally-finite categories.
I don't believe there are answers for enriched categories yet.