This "tensor product" is also known as the weighted colimit in enriched category theory. The short answer is that all the isomorphisms you are interested in always exist, provided the objects you are interested in also exist – it's just a matter of choosing the right definitions.

In general, given a complete symmetric monoidal closed category $\mathcal{V}$, a $\mathcal{V}$-enriched category $\mathcal{A}$, a small $\mathcal{V}$-enriched category $\mathcal{C}$, and $\mathcal{V}$-enriched functors $F : \mathcal{C}^\mathrm{op} \to \mathcal{V}$ (a.k.a. a right $\mathcal{C}$-module) and $G : \mathcal{C} \to \mathcal{A}$, the **$\mathcal{V}$-enriched weighted colimit** $F \star G$ is an object in $\mathcal{A}$ equipped with isomorphisms
$$[\mathcal{C}^\mathrm{op}, \mathcal{V}](F, \mathcal{A} (G, a)) \cong \mathcal{A} (F \star G, a)$$
that are $\mathcal{V}$-enriched natural in $a$.

Dually, given $\mathcal{V}$-enriched functors $F : \mathcal{C} \to \mathcal{V}$ and $G : \mathcal{C} \to \mathcal{A}$, the **$\mathcal{V}$-enriched weighted limit** $\{ F, G \}$ is an object in $\mathcal{A}$ equipped with isomorphisms
$$[\mathcal{C}, \mathcal{V}](F, \mathcal{A} (a, G)) \cong \mathcal{A} (a, \{ F, G \})$$
that are $\mathcal{V}$-enriched natural in $a$.

The $\mathcal{V}$-enriched Yoneda lemma yields:

- If $F$ is $\mathcal{C} (-, c)$, then $F \star G$ exists and is isomorphic to $G c$.
- If $F$ is $\mathcal{C} (c, -)$, then $\{ F, G \}$ exists and is isomorphic to $G c$.

In other words, the weighted colimit $F \star G$ (resp. the weighted limit $\{ F, G \}$) always exists if $F$ is a representable $\mathcal{V}$-enriched functor $\mathcal{C}^\mathrm{op} \to \mathcal{V}$ (resp. $\mathcal{C} \to \mathcal{V}$). A standard adjointness argument can be used to show that ${-} \star G$ (resp. $\{ {-}, G \}$) preserves colimits (resp. limits) in a strong sense; so, for instance, if $F$ is a conical colimit of a finite diagram of representables and $\mathcal{A}$ has conical colimits of finite diagrams, then $F \star G$ exists and is the conical colimit of the obvious diagram.

Very explicitly: let $C : \mathcal{I} \to \mathcal{C}$ be a $\mathcal{V}$-enriched functor and let $F : \mathcal{C}^\mathrm{op} \to \mathcal{V}$ be $\varinjlim \mathcal{C} (-, C)$; then $F \star G$ exists in $\mathcal{A}$ if and only if $\varinjlim G C$ exists in $\mathcal{A}$, and they are isomorphic. Indeed:
\begin{align}
\mathcal{A} (F \star G, a)
& \cong [\mathcal{C}^\mathrm{op}, \mathcal{V}](F, \mathcal{A} (G, a)) \\
& \cong [\mathcal{C}^\mathrm{op}, \mathcal{V}](\varinjlim \mathcal{C}(-, C), \mathcal{A} (G, a)) \\
& \cong \varprojlim [\mathcal{C}^\mathrm{op}, \mathcal{V}](\mathcal{C}(-, C), \mathcal{A} (G, a)) \\
& \cong \varprojlim \mathcal{A} (G C, a) \\
& \cong \mathcal{A} (\varinjlim G C, a)
\end{align}

Of course, there are also "two-sided" or "parametrised" versions of these: so given a $\mathcal{V}$-enriched functors $F : \mathcal{C}^\mathrm{op} \otimes \mathcal{D} \to \mathcal{V}$ (a.k.a. a bimodule) and $G : \mathcal{C} \to \mathcal{A}$ we can define $F \star_\mathcal{C} G$ to be a $\mathcal{V}$-enriched functor $\mathcal{D} \to \mathcal{A}$ equipped with isomorphisms
$$[\mathcal{C}^\mathrm{op} \otimes \mathcal{D}, \mathcal{V}](F, \mathcal{A}(G, H)) \cong [\mathcal{D}, \mathcal{A}](F \star_\mathcal{C} G, H)$$
that are $\mathcal{V}$-enriched natural in $H$; and similarly, $\{ F, H \}^\mathcal{D}$ is in turn defined to be a $\mathcal{V}$-enriched functor $\mathcal{C}^\mathrm{op} \to \mathcal{A}$ equipped with isomorphisms
$$[\mathcal{C}^\mathrm{op} \otimes \mathcal{D}, \mathcal{V}](F, \mathcal{A}(G, H)) \cong [\mathcal{C}^\mathrm{op}, \mathcal{A}](G, \{ F, H \}^\mathcal{D})$$
that are $\mathcal{V}$-enriched natural in $G$. In particular, when $F \star_\mathcal{C} G$ and $\{ F, H \}^\mathcal{D}$ both exist, then:
$$[\mathcal{D}, \mathcal{A}](F \star_\mathcal{C} G, H) \cong [\mathcal{C}^\mathrm{op}, \mathcal{A}](G, \{ F, H \}^\mathcal{D})$$

For your situation, take $\mathcal{V}$ to be the category of $k$-modules. In this special case, there is no difference between "natural" and "enriched natural", nor between "conical colimit" and "(ordinary) colimit". The question of existence is easy here: a functor $F : \mathcal{C}^\mathrm{op} \to \operatorname{Mod} k$ is finitely presented if and only if it can be obtained from representable functors by iterated colimits of finite diagrams. (In fact, two steps suffice: first form a finite direct sum of representable functors, then take a cokernel.)