It's easiest to understand this question by relating $V$-categories to $V$-graphs. A $V$-graph $G$ is given by a set of objects $\mathrm{ob} G$ together with an object of $V$, denoted by $G(x,y)$, for every $x,y\in \mathrm{ob} G$.
As in the case of $V=\mathrm{Set}$, limits of $V$-categories are created by the forgetful functor into $V$-graphs. This means that the object set and the homs in a limit of $V$-categories are given by the corresponding limits of sets and of hom-objects in $V$, respectively. Coproducts of $V$-categories are just disjoint unions, so it's only coequalizers that really present some difficulties. That said, they really present some difficulties! See the following (freely available) paper of Wolff for the full construction, which proves along the way that $V$-categories are monadic over $V$-graphs. Wolff's paper
Frequently, the enriching category $V$ is in fact locally presentable, not just complete and cocomplete. In fact this is pretty much always the case unless $V=\mathrm{Top}$. In that case it may be shown that $V$-Cat is also locally presentable, which gives a higher-level proof that the latter is cocomplete. It was proved much later than Wolff's paper by Lack and Kelly that the following hold, if $V$ is cocomplete and tensors preserve colimits-for instance $V$ could be closed, but this is not required, and $V$ needn't be symmetric:
- $V$-Cat is in fact finitarily monadic over $V$-Graph, that is, the forgetful functor's left adjoint giving the free $V$-category on a graph preserves filtered colimits.
- $V$-Graph is locally $\lambda$-presentable when $V$ is so. This requires direct arguments at least as complex as Wolff's, though couched in a nice general formalism of $V$-matrices.
- By general nonsense, we conclude from the previous two points that $V$-cat is $\lambda$-locally presentable when $V$ is so, and in particular must be cocomplete.
Neither of these papers produces a practically usable algorithm for computing colimits of $V$-categories, but this is unavoidable-for $V=$Set, considering $V$-categories with one object and focusing on coequalizers of maps between free objects, we have recovered the question of computing a monoid from generators and relations, which is known to be generally undecidable.
I follow Kelly [Basic concepts of enriched category theory].
Q1. In general, no; in practice, yes.
By definition, we have an equaliser diagram
$$\int_{i : \mathcal{I}} \mathcal{A} (F (i), G (i)) \rightarrow \prod_{i \in \operatorname{ob} \mathcal{I}} \mathcal{A} (F (i), G (i)) \rightrightarrows \prod_{(i, j) \in (\operatorname{ob} \mathcal{I})^2} \mathcal{V} (\mathcal{I} (i, j), \mathcal{A} (F (i), G (j)))$$
and applying the underlying set functor preserves limits, so we get an equaliser diagram of sets:
$$\left( \int_{i : \mathcal{I}} \mathcal{A} (F (i), G (i)) \right)_0 \rightarrow \prod_{i \in \operatorname{ob} \mathcal{I}} \mathcal{A}_0 (F (i), G (i)) \rightrightarrows \prod_{(i, j) \in (\operatorname{ob} \mathcal{I})^2} \mathcal{V}_0 (\mathcal{I} (i, j), \mathcal{A} (F (i), G (j)))$$
This is almost – but not quite – the definition of $\int_{i : \mathcal{I}_0} \mathcal{A}_0 (F (i), G (i))$.
For the latter we would have $\prod_{(i, j) \in (\operatorname{ob} \mathcal{I})^2} \textbf{Set} (\mathcal{I}_0 (i, j), \mathcal{A}_0 (F (i), G (j)))$ instead.
For every $V$ and $W$ in $\mathcal{V}$, the action of the underlying set functor $\mathcal{V}_0 \to \textbf{Set}$ is a map
$$\mathcal{V}_0 (V, W) \longrightarrow \textbf{Set} (V_0, W_0)$$
so we get a canonical map:
$$\prod_{(i, j) \in (\operatorname{ob} \mathcal{I})^2} \mathcal{V}_0 (\mathcal{I} (i, j), \mathcal{A} (F (i), G (j))) \longrightarrow \prod_{(i, j) \in (\operatorname{ob} \mathcal{I})^2} \textbf{Set} (\mathcal{I}_0 (i, j), \mathcal{A}_0 (F (i), G (j))) \tag{†}$$
This induces a comparison map
$$\left( \int_{i : \mathcal{I}} \mathcal{A} (F (i), G (i)) \right)_0 \longrightarrow \int_{i : \mathcal{I}_0} \mathcal{A}_0 (F (i), G (i)) \tag{‡}$$
and it is easy to see that (‡) is a bijection if (†) is an injection.
This happens if the underlying set functor $\mathcal{V}_0 \to \textbf{Set}$ is faithful, which is usually the case in practice.
(The main exception is when $\mathcal{V}$ is a category of multisorted structures, e.g. simplicial sets or chain complexes.)
Q2.
Yes.
When $V$ is a coproduct of $X$ copies of the monoidal unit, then the underlying set functor gives a bijection $\mathcal{V}_0 (V, W) \to \textbf{Set} (X, W)$.
Thus, if $\mathcal{I}$ is the free $\mathcal{V}$-category on an ordinary category, then (†) is a bijection, so (‡) is also a bijection.
(But we could have deduced this directly from the universal property of free $\mathcal{V}$-categories!)
Q3.
Yes.
Suppose $F$ and $G$ are constant functors, with values $A$ and $B$ respectively.
You can directly verify that the diagonal morphism $\mathcal{A} (A, B) \to \prod_{i \in \operatorname{ob} \mathcal{I}} \mathcal{A} (F (i), G (i))$ equalises $\prod_{i \in \operatorname{ob} \mathcal{I}} \mathcal{A} (F (i), G (i)) \rightrightarrows \prod_{(i, j) \in (\operatorname{ob} \mathcal{I})^2} \mathcal{V} (\mathcal{I} (i, j), \mathcal{A} (F (i), G (j)))$ and so we have the desired morphism $\mathcal{A} (A, B) \to \int_{i : \mathcal{I}} \mathcal{A} (F (i), G (i))$.
(Again, you could instead use the universal property of free $\mathcal{V}$-categories here.
The point is that a $\mathcal{V}$-functor $\mathcal{A} \to [\mathcal{I}, \mathcal{A}]$ is essentially the same thing as a $\mathcal{V}$-functor $\mathcal{I} \to [\mathcal{A}, \mathcal{A}]$, and when $\mathcal{I}$ is free we can just take the constant functor with value $\textrm{id}_\mathcal{A}$.)
Best Answer
It's basically necessary that $V$ be complete and cocomplete (for instance, it's definitely necessary if $V$ admits any absorbing object for its monoidal product), so let's assume that.
For limits of $V$-categories to exist, this is sufficient-in fact only completeness is necessary. The limit of a diagram $D:J\to V-\mathrm{Cat}$ is the $V$-category with objects the limit of $\mathrm{ob} D(j)$ taken in $\mathrm{Set}$, while the morphisms $\mathrm{lim} D((a_j),(b_j))$ are just given by the limit $\mathrm{lim} D(j)(a_j,b_j)$ taken in $V$. The composition operation uses the canonical map $$\mathrm{lim} D(j)(a_j,b_j)\otimes \mathrm{lim} D(j)(b_j,c_j)\to \mathrm{lim}\left(D(j)(a_j,b_j)\otimes D(j)(b_j,c_j)\right).$$ Since this canonical map does not exist in the case of colimits, the latter are harder.
I don't have either a proof or a counterexample available in the case that $V$ is merely cocomplete, but this is easy to show cocompleteness in the case of simplicially enriched categories, which you say is your main interest. Indeed, it's immediate that the category of simplicial objects in categories, that is, the functor category $\mathrm{Cat}^{\Delta^{\mathrm{op}}}$, is cocomplete, since $\mathrm{Cat}$ is. And simplicially enriched categories may be identified with the subcategory of $\mathrm{Cat}^{\Delta^{\mathrm{op}}}$ such that the simplicial set $\mathrm{ob} C_\bullet$ of objects is discrete. This property is preserved by colimits, which gives colimits for simplicial categories.
This argument can be generalized by replacing $\Delta^{\mathrm{op}}$ with any small category $A$ and the condition that $\mathrm{ob} C_\bullet$ be discrete with the condition that it be a constant presheaf (of sets) on $A$. Then the point is that the "constant presheaf" functor is fully faithful with a right adjoint. For most other $V$ of interest, we will not be able to embed $\mathrm{Set}$ fully faithfully via a left adjoint, so we cannot reduce the question of colimits of categories enriched in $V$ to the easier question of colimits of categories internal to $V$ like this. Cocompleteness should still hold at least if $V$ is monoidally locally presentable, which will make $V$-categories accessible monoidal over $V$-graphs and thus locally presentable. But I don't have a reference for this argument.