The definition is careless. Although relying solely on a supremum might get me into trouble, I think everything will work properly with a limsup (Besicovitch uses these upper means in his own investigation of the almost periodics). Then we look at locally $p$-integrable functions finite under this seminorm (a real seminorm this time), which will be a Banach space after modding out and completing (if necessary).

Of course. The $B^p$ functions have all the nice properties, like plenty of $epsilon$-translation numbers, which make them almost periodic. The $M^p$ functions almost just seem like a nice place to house the $B^p$.

On a general locally compact abelian group, the Fourier transform acts $\widehat{\cdot} : L^1(G,\mu) \to \text{C}_\infty(\widehat{G})$ where $\mu$ is a Haar measure on $G$, and Pontryagin duality exists between $G$ and $\widehat{G}$ (so that $G\cong \widehat{\widehat{G}}$). In general we have $\widehat{f*g} = \widehat{f}\cdot \widehat{g}$. If $\widehat{f},\widehat{g} \in L^1(\widehat{G},\nu)$ (where $\nu$ is the dual measure to $\mu$), then the duality allows us to say that $\widehat{\widehat{f}*\widehat{g}} = f\cdot g$. That might get you part of the way to an answer.