Let's start with the cartesian monoidal case, since it is the easiest one to understand. Recall that the Yoneda embedding is left exact, so if we have a monoid $M$ in a category $\mathcal{C}$, then $\mathcal{C}(-, M)$ is automatically a monoid in the presheaf topos $[\mathcal{C}^\textrm{op}, \textbf{Set}]$, and for _any_ object $X$ in $\mathcal{C}$, the hom-set $\mathcal{C}(X, M)$ is an ordinary monoid in $\textbf{Set}$.

More explicitly, if the structural data of $M$ are $e : 1 \to M$ and $m : M \times M \to M$, we get induced maps $e_* : \mathcal{C}(X, 1) \to \mathcal{C}(X, M)$ and $m_* : \mathcal{C}(X, M \times M) \to \mathcal{C}(X, M)$; but left exactness means that $\mathcal{C}(X, 1) \cong 1$ and $\mathcal{C}(X, M \times M) \cong \mathcal{C}(X, M) \times \mathcal{C}(X, M)$, so this indeed induces the structure of a monoid on $\mathcal{C}(X, M)$.

Now let $\mathcal{C}$ be a monoidal category. I will pretend it is strict monoidal. Let $M$ be a monoid in $\mathcal{C}$. As before, we get maps $\mathcal{C}(X, I) \to \mathcal{C}(X, M)$ and $\mathcal{C}(X, M \otimes M) \to \mathcal{C}(X, M)$, but unfortunately there is no obvious map $1 \to \mathcal{C}(X, I)$ or $\mathcal{C}(X, M) \times \mathcal{C}(X, M) \to \mathcal{C}(X, M \otimes M)$; that is to say, $\mathcal{C}(X, -)$ is not automatically a lax monoidal functor.

But hope is not lost yet: it turns out $\mathcal{C}(I, -)$ _is_ a lax monoidal functor! There is an obvious map $1 \to \mathcal{C}(I, I)$ (namely the constant map with value $\textrm{id}_I$) and a natural map $\mathcal{C}(I, Y) \times \mathcal{C}(I, Z) \to \mathcal{C}(I, Y \otimes Z)$ given by $(f, g) \mapsto f \otimes g$ (and suppressing the isomorphism $I \otimes I \cong I$, as previously mentioned), and it is easy to check that this does indeed give $\mathcal{C}(I, -)$ the structure of a lax monoidal functor.

Finally, it is a well-known fact that lax monoidal functors carry monoids to monoids. Let's prove this now. Suppose $F : \mathcal{C} \to \mathcal{D}$ is a lax monoidal functor and $M$ is a monoid in $\mathcal{C}$. Then we get a morphism $I_\mathcal{D} \to F M$ by composing $I_\mathcal{D} \to F I_\mathcal{C}$ with $F e : F I_\mathcal{C} \to F M$, and a morphism $F M \otimes_\mathcal{D} F M \to F M$ by composing $F M \otimes_\mathcal{D} F M \to F (M \otimes_\mathcal{C} M)$ with $F m : F (M \otimes_\mathcal{C} M) \to F M$. The coherence axioms for lax monoidal functors imply that $F M$ together with these structural data satisfy the monoid axioms. For example, to check that the right unit axiom holds, we must show that the composite
$$F M \to F M \otimes_\mathcal{D} I_\mathcal{D} \to F M \otimes_\mathcal{D} F M  \to F (M \otimes_\mathcal{C} M ) \to F M$$ 
is equal to the identity, but this is equal to the composite
$$F M \to F M \otimes_\mathcal{D} I_\mathcal{D} \to F M \otimes_\mathcal{D} F I_\mathcal{C} \to F (M \otimes_\mathcal{C} I_\mathcal{C}) \to F (M \otimes_\mathcal{C} M ) \to F M$$ 
by naturality of $F Y \otimes_\mathcal{D} F Z \to F (Y \otimes_\mathcal{C} Z)$, and this is equal to the identity by the right unit axiom for $M$ together with the coherence axiom for $I_\mathcal{D} \to F I_\mathcal{C}$.

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Step 2 of your question is also related to the Yoneda embedding, but in a more subtle way. In effect, it is externalising the internal monoid $M$ by considering its left self-action. This is basically an appeal to the monoid version of Cayley's theorem, which is sometimes considered a special case of the Yoneda embedding.

Personally, I don't see a connection with the Kleisli category construction. Perhaps one can regard the "embedding" of $\mathcal{C}(I, X)$ into $\mathcal{C}(X, X)$ as an instance of a "funny composition", but I don't think there's anything deeper than that.