Cilve, your first instance shows that we have to be careful when moving from $\mathcal{D}$ to $[\mathcal{C}, \mathcal{D}]$ --- you should easily find categories $\mathcal{C}, \mathcal{D}$ such that $\mathcal{D}$ is cartesian closed, but the functor category $[\mathcal{C}, \mathcal{D}]$ is not.
The other examples follow more-or-less from a suitable version of the Yoneda lemma. I will show you how to apply Yoneda lemma to get limits/colimits in a functor category. Let $\mathcal{X}, \mathcal{D}$ be small categories. There exists their cotensor (exponent) $\mathcal{D}^\mathcal{X}$ together with the diagonal functor:
$$\mathcal{D} \overset{\Delta}{\rightarrow} \mathcal{D}^\mathcal{X}$$
given as the transposition of the cartesian projection $\pi \colon \mathcal{D} \times \mathcal{X} \rightarrow \mathcal{D}$. The limit functor $\mathit{lim}$ is defined as the right adjoint to the diagonal, and the colimit $\mathit{colim}$ is defined as the left adjoint to the diagonal.
The whole idea is to apply the 2-Yoneda functor to the above diagram. We have $$\hom(-, \mathcal{D}^\mathcal{X}) \approx \hom(-, \mathcal{D})^{\mathcal{X}}$$
by the definition of the cotensor. Therefore the above diagram is mapped to the diagram:
$$\hom(-, \mathcal{D}) \rightarrow \hom(-, \mathcal{D})^\mathcal{X}$$
Since adjunctions are equationally defined, they are preserved by any 2-functor, and particularly by 2-Yoneda. This means, that the above transformation has right/left adjoint transformation provided $\Delta$ has. But a transformation that has left/right adjoint, has left/right adjoint on its each componenet $\mathcal{C}$. Thus:
$$\hom(\mathcal{C}, \mathcal{D}) \rightarrow \hom(\mathcal{C}, \mathcal{D})^\mathcal{X}$$
has right/left adjoint if $\mathcal{D}$ has $\mathcal{X}$-indexed limits/colimits.
We may also see what would go wrong if one tried to apply the above strategy to show that cartesian closedness is inherited by functor categories.
Let us recall that a category $\mathcal{D}$ is cartesian closed if for every global element $x \colon 1 \rightarrow \mathcal{D}$ the canonical functor:
$$\mathcal{D} \approx \mathcal{D} \times 1 \overset{\mathit{id}\times x}\rightarrow \mathcal{D} \times \mathcal{D} \overset{\times_\mathcal{D}}{\rightarrow} \mathcal{D}$$
has right adjoint, where $\mathcal{D} \times \mathcal{D} \overset{\times_\mathcal{D}}{\rightarrow} \mathcal{D}$ is the internal cartesian product functor in $\mathcal{D}$.
Just like before, we may apply to our diagram the 2-Yoneda functor obtaining:
$$\hom(-, \mathcal{D}) \approx \hom(-, \mathcal{D}) \times 1 \overset{\mathit{id}\times \hom(-, x)}\rightarrow \hom(-, \mathcal{D}) \times \hom(-, \mathcal{D}) \overset{\times_{\hom(-, \mathcal{D})}}{\rightarrow} \mathcal{D}$$
and conclude that this transformation has right adjoint iff the former has. However, the terminal object $1$ is not a (2-)generator in $\mathbf{Cat}^{\mathbf{cat}^{op}}$, thus the adjunctions do not give a good characterisation of internally cartesian closed objects in that category. Particularly, $\hom(-, \mathcal{D}) \colon \mathbf{cat}^{op} \rightarrow \mathbf{Cat}$ may not be a cartesian closed (2-)fibration, and one may expect existence of exponents in a fibre $\hom(\mathcal{C}, \mathcal{D})$ only on "constant" objects induced by global sections $\hom(\mathcal{C}, x) \colon 1 \rightarrow \hom(\mathcal{C}, \mathcal{D})$.
