When you take the monoidal category to be Set (which is locally small) with the cartesian product, you recover the usual Yoneda lemma. If you're really interested in the size condition, perhaps the right place to look is into Yoneda structures, which attempt to axiomatise this structure and have to pay close attention to size.

Yes, there is an enriched Yoneda lemma: see this question, or this nLab page.

A similar discussion may be found in Representability relative to a doctrine.

My understanding is that it is bicomplete (which follows from the definition of CFS). However, I should note that there is an issue with terminology, in that you'll find it stated (e.g. in this answer) that the Isbell envelope may not have small limits or colimits: but this instead refers to the construction arising from the Isbell adjunction (the distinction is discussed here). I don't know whether the functor you mention has adjoints, either.

Day's reflection theorem tells us that the LFP category $\mathscr C$ is cartesian-closed iff the reflector is cartesian: how can we characterise the corresponding category of finitely presentable objects for $\mathscr C$ using this?

Thank you, both of those references are perfect!

Thanks. The construction via a (2-categorical) presentation was the one I had in mind, but that taking pseudomorphisms instead of strict morphisms suffices to obtain exactly the continuous functors seems a little subtle, which is why I would have liked to seen it proven. Additionally, if this construction works for, say, finite completion, I see no reason it would not also work for small completion (with an appropriate notion of large presentation), but several people have expressed to me they would be surprised if the small cocompletion pseudomonad could be strictified into a 2-monad.

Thanks, this is a nice example. I think simply knowing that there isn't such an elegant characterisation for (1) is enough to answer that part of my question. I had hoped there would be a condition not much more complex than that for (2), but suspected it might not be the case.

Ah, perhaps the answer is that, if $\mathbf{Ind}_\kappa(\mathcal C)$ is cocomplete, then there is a category $\mathcal C'$ with $\kappa$-small colimits such that $\mathbf{Ind}_\kappa(\mathcal C) ≃ \mathbf{Ind}_\kappa(\mathcal C')$, but $\mathcal C$ itself may not be.

To clarify where my confusion is coming from, I was basing my original statement of Theorem 5.5(ii) of A classification of accessible categories (which I see I must have misunderstood). That paper does not mention idempotents, so I had not appreciated their importance here.

Sorry, I realise I omitted a critical negation in my previous comment. Thank you for your patience! Suppose $\mathbf{Ind}_\kappa(\mathcal C)$ is cocomplete. If idempotents in $\mathcal C$ split, then we know that $\mathcal C$ has $\kappa$-small colimits. However, if idempotents in $\mathcal C$ do not split, then $\mathcal C$ cannot have $\kappa$-small colimits, or otherwise its idempotents would split. Is there a specific example of a small category in which idempotents do not split, but whose $\mathbf{Ind}_\kappa$-completion is cocomplete?