No a answere but a specification and description of the issue (too long for a comment).
The free cocompletion property of $Ind(\mathcal{C})$ dont use the hypothesis "$\mathcal{C}$ has finite colimits". ANyway $Ind(\mathcal{C})$ is equivalent to the full subcategory $P_1(\mathcal{C})\subset \mathcal{C}^>$ of presheves that are coimits of a small, filtred diagram of representable. In this way the inclusion $i$ corresponds to the yoneda inclusion $h_-: \mathcal{C}\to P_1(\mathcal{C})$ and it preserves limit, and finite colimits.
The inclusion $P_1(\mathcal{C})\subset \mathcal{C}^>$ create (small) filter colimits (if $P=\varinjlim_{i\in I}P_i$ is a filtrant colimit with $O_i\in P_1(P_1(\mathcal{C})$ then combining the comma categories $P_1(\mathcal{C})\downarrow P_i$ we
make a filtrant (small) diagram of $P_1(\mathcal{C})\downarrow P_i$
Is $F: \mathcal{C}\to \mathcal{D}$ where $\mathcal{D}$ as filter colimits we have a (iso)unique extentions $F': P_1(\mathcal{C})\to \mathcal{D}$ with $F'(P):=\varinjlim_{\ (X, x)\in \mathcal{C}\downarrow P} F(X)$, if $P$ has the Ind-representation $(X_i)_{i\in I}$ i.e. $P=\varinjlim_i h_{X_i}$ then the diagram of the $h_{X_i}$ is a final diagram on the comma category $\mathcal{C}\downarrow P$ then $F'(P):=\varinjlim_i F(X_i)$
If $\mathcal{C}$ has finite colimits $P_1(\mathcal{C})\cong Cart(\mathcal{C}^{op}, Set)$ the latter is the the category cartesians presheaves i.e. that maps finite colimits of $\mathcal{C}$ to finite limits in $Set$, of course the embedding $Cart(\mathcal{C}^{op}, Set)\subset \mathcal{C}^>$ create limits .
All above as a dual version for $Proj(\mathcal{C}):=(Ind(\mathcal{C}^{op}))^{op}$, it is a free completion of $\mathcal{C}$, and if $\mathcal{C}$ has finite limits $Proj(\mathcal{C})$ is equivalent to $Cart(\mathcal{C}, Set)^{op}$ (dual to the category of copresheaves that preserving finite limits) it has (small) colimits and we have the embedding $\iota:=(h^-)^{op}: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op} $.
Now the inclusion $h_-: \mathcal{C}\to Cart(\mathcal{C}^{op}, Set)$ and $\iota: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op}$ induce for the universal properties of these two completions the functors $U: Cart(\mathcal{C}, Set)^{op}\to Cart(\mathcal{C}^{op}, Set)$ with $U(Q)=\varprojlim_{\ h^Y\to Q} h_Y$, and
$F: Cart(\mathcal{C}^{op}, Set)\to Cart(\mathcal{C}, Set)^{op}$ with
$F(P)=\varprojlim_{\ h_X\to P} h^X$ (the limit is in $\mathcal{C}^{<}$ the copresheaves category)
we have the natural isomorphisms:
$Cart(\mathcal{C}, Set)^{op}(F(P), Q)=Cart(\mathcal{C}, Set)(Q, F(P))=$
$\mathcal{C}^{<}(\varinjlim_{\ h^Y\to Q}h^Y, \varprojlim_{\ h_X\to P}h^X)=$
$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^<(h^Y, h^X)\cong$
$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^>(h_X, h_Y)\cong$
$\mathcal{C}^>(\varinjlim_{\ h_X\to P} h_X, \varprojlim_{h^Y\to Q}h_Y) \cong$
$Cart(\mathcal{C}^{op}, Set)(P, U(Q))$
then $U$ is a adjoint to $F$.
So, the question appeared more subtle than I initially thoughts so I have written a short paper with more references and the details of what I'm going to say below. It is available here.
Here is the summary:
First, Makkai's theorem cited by Ivan is indeed false. Building on Ben Wieland's comment (thank you very much for this!), one get the following counter-example.
Let $I=B(\mathbb{Z}/2\mathbb{Z})$ the category with one object $*$ and one non-trivial automorphism satisfying $f^2=1$ and let $C$ be the subcategory of $I \times \omega$ containing all the objects, and all the arrows $(*,n) \to (*,m)$ when $n < m$ but only the identity when $n = m$.
In particular, functors $I \to C$ are all constant, as $C$ has no non-trivial endomorphisms, but there is a non-trivial functor $I \to Ind(C)$: indeed the colimits of the chain of the $(*,n)$ is in the Ind completion, and as a presheaf, it is the pullback along the projection $C \to I$ of the unique representable presheaf, so it comes with a $\mathbb{Z}_2$-action and hence is a non-trivial functor $ I \to Ind(C)$, which doesn't belong to $Ind(C^I)$.
As $C$ has no non-trivial endomorphisms it is Cauchy complete, hence $C$ identifies with the full subcategory of $\omega$-presentable objects of $Ind(C)$ and hence the exponential $Ind(C)^I$ is a counter-example to Makkai's theorem.
Now this construction can be generalized to any category and any ordinal, and this leads to the following theorem:
Theorem: Let $I$ be a category and $\kappa$ be a regular cardinal. The following condition are equivalent.
- $I$ is $\kappa$-small, has no non-identity endomorphisms, and its posetal relfection is well-founded.
- for all category $C$, the functor $Ind_\kappa(C^I) \to Ind_\kappa(C)^I$ is an equivalence.
- for all $\kappa$-accessible category $A$, the category $A^I$ is $\kappa$-accessible with its $\kappa$-presentable objects being the functors from $I$ to $\kappa$-presentable objects of $A$.
In particular, one recovers Lurie's version of the theorem (I mean for 1-categories), as well as Meyer's result mentioned by Peter Haine. Also Makkai theorem is false even for uncountable $\kappa$. But this isn't the end of the story. If we are interested in category with colimits, then everything works more nicely and we get a positive answer to my questions:
Theorem: Let $I$ be a category and $\kappa$ be a regular cardinal. The following conditions are equivalent.
- $I$ is $\kappa$-small.
- for all category $C$ with $\kappa$-small colimits, the functor $Ind_\kappa(C^I) \to Ind_\kappa(C)^I$ is an equivalence.
- for all locally $\kappa$-presentable category $A$, the category $A^I$ is locally $\kappa$-presentable, with its $\kappa$-presentable objects being the functors from $I$ to $\kappa$-presentable objects of $A$.
The proofs of these and additional details are in the paper linked above.
It seems Makkai's theorem was used in a few places in the literature - but from what I can tell, all uses I've seen are covered by these two theorems.
Best Answer
A positive answer to your first question can be seen as a consequence of Lemma 2.5 of On continuity of accessible functors. Following the notation of the Lemma, you should take $\beta$ to be the maximum between the the number of objects and the number of morphisms of your $\mathcal C$, and $\gamma=\aleph_0$. Then a $\gamma$-flat functor is simply a flat functor; while $\mathbf{Set}_\beta$ is the category of sets with less than $\beta$ elements, which is contained in $\mathbf{FinSet}$ since $\beta$ is finite.
Then, the fact that $Ind(\mathcal C)$ coincides with tha Cauchy completion of $\mathcal C$ is a consequence of 2.6 and 2.8 of the same paper.