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 $I_\mathcal{D} \to F I_\mathcal{C}$.

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.

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 $I_\mathcal{D} \to F I_\mathcal{C}$.

More generally, $\mathcal{C}(X, -)$ is lax monoidal whenever $X$ is a comonoid. This recovers the result of the first paragraph, since in a cartesian monoidal category, every object is a comonoid in a unique way.

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.