Let $(C, \otimes, I)$ be a symmetric monoidal category (for example, modules over a commutative ring), and let $M$ be a dualizable object with dual $M^\vee$. Then $\_ \otimes M$ is left adjoint to $\mathcal{H}om(M, \_) \cong M^\vee\otimes\_$. Applying the same logic to $M^\vee$, we have that $\_\otimes M^\vee$ is left adjoint to $M\otimes \_$ since $({M^\vee})^\vee\cong M$. Furthermore, since $(C, \otimes)$ is symmetric monoidal, then the functor $\_\otimes M$ is equivalent to the functor $M\otimes \_$ and similarly for $M^\vee$.
In summary, we have that $M\otimes \_$ is both left and right adjoint to $M^\vee\otimes\_ ≅\_ \otimes M^\vee$ (and $\_ \otimes M^\vee$ is both left and right adjoint to $\_ \otimes M$).
Question 1: what are some other examples of such situations, i.e. functor $F: C→D$ and $G: D\rightarrow C$ such that $F$ is both left and right adjoint to $G$ (and $G$ is left and right adjoint to $F$) but these are not equivalences.
Question 2: What happens when $(C, \otimes)$ is not symmetric monoidal?
I believe that in this case one must distinguish between a left and right dual for $M$ and the resulting functors are left/right adjoints respectively.
Question 3: Can a similar situation occur when $M$ is a $R-S$-bimodule, for non-commutative rings $R,S$ and the adjoint functors are $\otimes_{R-mod} M: mod-R \rightarrow mod-S$ and $Hom_{mod-S}(M, \_): mod-S\rightarrow mod-R$.
I am confused about what exactly the dualizability condition is.