Let $\epsilon^{A}_B$ be the map $\eta_{\frac{B}{A},A,B}^{-1}(1_{\frac{B}{A}}) $, or in other words, let $\epsilon^{A}$ be the counit of the adjunction $? \otimes_A\dashv \frac{??}{A}$, and similarly for $B,C$. Then the inverse of $\eta_{A,B,C}$ is the map $g\mapsto \epsilon^{B}_C \circ (g\otimes B)$. Moreover, the inner composition
$$\frac{C}{B}\otimes\frac{B}{A}\to\frac{C}{A}$$
is the image of the composite
$$\frac{C}{B}\otimes\frac{B}{A}\otimes A\stackrel{\frac{C}{B}\otimes \epsilon_{B}^{A}}{\longrightarrow} \frac{C}{B}\otimes B \stackrel{ \epsilon_{C}^B}{\longrightarrow} C$$
under the bijection $\eta_{\frac{C}{B}\otimes\frac{B}{A},A,C}$.
As a consequence, and because of the naturality of $\eta$, we find that
\begin{align}\bullet_{A,B,C}\circ (\widehat{\psi}\otimes\widehat{\varphi}) & = \eta_{\frac{C}{B}\otimes\frac{B}{A},A,C}\left(\epsilon^B_C\circ \left(\frac{C}{B}\otimes\epsilon^{A}_B\right) \right) \circ (\widehat{\psi}\otimes \widehat{\varphi})\\
& = \eta_{I\otimes I,A,C}\left(\epsilon^B_C\circ \left(\frac{C}{B}\otimes\epsilon^{A}_B\right) \circ (\widehat{\psi}\otimes \widehat{\varphi}\otimes A)\right) \\
& = \eta_{I\otimes I,A,C}\left(\epsilon^B_C\circ \left(\widehat{\psi}\otimes (\epsilon^{A}_B \circ ( \widehat{\varphi}\otimes A))\right) \right) \\
& = \eta_{I\otimes I,A,C}\left(\epsilon^B_C \circ \left(\widehat{\psi}\otimes (\varphi\circ \lambda_A) \right) \right) \\ &
= \eta_{I,A,C}\left(\epsilon^B_C \circ (\widehat{\psi}\otimes B) \circ (I\otimes (\varphi\circ \lambda_A)) \right) \\
& = \eta_{I\otimes I,A,C}\left(\psi\circ \lambda_B \circ (I\otimes (\varphi\circ \lambda_A)) \right)\\
& = \eta_{I\otimes I,A,C}\left(\psi\circ \varphi\circ \lambda_A \circ \lambda_{I\otimes A} \right) \\
& = \eta_{I\otimes I,A,C}\left((\psi\circ \varphi)\circ \lambda_A \circ (\lambda_{I}\otimes A) \right)\\
& = \eta_{I,A,C}\left((\psi\circ \varphi)\circ \lambda_A \right)\circ \lambda_{I} \\ & = \widehat{\psi\circ \varphi} \circ \lambda_I\end{align}
(with some associators missing, but it should work).
There seems to be many different questions asked here. I'll answer some of them.
The free/forgetful adjunction that you want looks like this:
Consider the bicategory $\textsf{Cat}$ of categories, functors and natural transformations. Consider the bicategory $\textsf{MCat}$ of monoidal categories, (strong) monoidal functors and monoidal natural transformations. There is an obvious 2-functor $\textsf{MCat} \to \textsf{Cat}$. The free monoidal category generated by a category is then the bi-adjunction. For the strict case just throw away all the natural transformations to get 1-categories. You may also define the monoidal category generated by a graph, or a set of objects, etc. in a similar way.
To address the $A \otimes B \to C$ generator problem, we would like to intertwine the process of generating things and stating new generators and equations. This is exactly the problem addressed by sketches. Traditionally we usually deal with freely generated limits/colimits. But this can be done with any 2-monad (in your case, you need the 2-monad that generates a free monoidal category from a category). See this for a treatment.
Concrete syntactic treatment (such as those you have linked) can also be done, it's basically just a huge inductive definition, quotiented by all the necessary equations. This is routinely done in type theory.
Best Answer
No. For instance, taking the category of sets with the Cartesian product, the unit object (a singleton) is not integral since it has no morphism to the empty set. More generally, if you take the category of sheaves (of sets, or of abelian groups if you want an abelian category say) on some site, both properties (1) and (2) will typically fail since a sheaf can have no global sections or not be generated by global sections. For instance, if you take the category of sheaves of sets on $\mathbb{R}$ and let $X$ be the sheaf given by $X(U)=\{*\}$ if $U\subseteq(0,1)$ and $X(U)=\emptyset$ otherwise, the two inclusions $X\to X\coprod X$ are distinct but $X$ has no morphisms from the unit object (which is the constant singleton sheaf).