Day convolution for bimonoidal categories Semiring categories, also called rig categories or bimonoidal categories, are pseudomonoids in the symmetric monoidal bicategory $(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})$¹. These are a categorification of semirings, the monoids in $(\mathsf{CMon},\otimes_{\mathbb{N}},\mathbb{N})$, and carry two monoidal structures, one additive and one multiplicative, with the multiplicative one being coherently bilinear over the additive monoidal structure. A great introduction for these is Johnson–Yau's Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic $K$-Theory.
Similarly, ring categories are pseudomonoids in $(\mathsf{2Ab},\otimes_{\mathbb{S}},\mathbb{S})$. They are a categorification of rings, the monoids in $(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z})$, and are more simply those semiring categories having (weak) additive inverses (i.e. for each object $A$ in a ring category, there exists an object $-A$ such that $A\oplus(-A)\cong\mathbf{0}_{\mathcal{C}}$ via a coherent isomorphism).
It is well-known that given a monoidal category $(\mathcal{C},\otimes,\mathbf{1}_{\mathcal{C}})$, we can use Day convolution to get a monoidal structure on presheaves on $\mathcal{C}$, obtaining a monoidal category $(\mathsf{PSh}(\mathcal{C}),\circledast,\mathsf{h}_{\mathbf{1}_{\mathcal{C}}})$.
Question 1. Given a semi/ring category $(\mathcal{C},\oplus,\otimes,\mathbf{1}_{\mathcal{C}},\mathbf{0}_{\mathcal{C}},\ldots)$, is the tuple $(\mathcal{C},\circledast^{\oplus},\circledast^{\otimes},\mathsf{h}_{\mathbf{1}_{C}},\mathsf{h}_{\mathbf{0}_{C}},\ldots)$ obtained by applying Day convolution to both monoidal structures also a semi/ring category?
Question 2. Day convolution gives a bijection
$$
\{\text{promonoidal structures on $\mathcal{C}$}\}
\cong
\{\text{biclosed monoidal structures on $\mathsf{PSh}(\mathcal{C}$})\}.
$$
Assuming the statement in question 1 holds, is there an analogue of this bijection for semi/ring categories?

¹More or less―for nonsymmetric bimonoidal categories, 19 of the 22 nonsymmetric bimonoidal category axioms of Johnson–Yau, Definition 2.1.2 hold. The exceptions are 2.1.13, 2.1.15, and 2.1.16.
 A: Regarding Q2: probably there is a way to avoid going deep into coherence conditions: instead of proving by hand the equivalence between promonoidal structures on $C$ and biclosed monoidal structures on $\hat C$, one can resort to a more conceptual pov.
What happens for pro/monoidal categories is that there is a pseudomonad $S$ on $\sf Cat$ with the property that $S$ lifts to a pseudomonad $\hat S$ on $\sf Prof$ (the Kleisli bicategory of $P=\hat{(-)} = [(-)^{op},{\sf Set}]$), and pseudo-$S$-algebra structures correspond to pseudo-$\hat S$-algebra structures (this is an equivalence of categories, in the appropriate sense; see here).
I believe a similar argument holds for every (almost every?) monad $S$ equipped with a distributive law over $P$ (the presheaf construction); this does not fall short from an equivalence
$$
\{S\text{-algebra structures on } PX\} \cong \{\hat S\text{-algebra structures on } X\}
$$ where $PX$ is regarded as an object of $\sf Cat$, and $X$ as an object of ${\sf Kl}(P)$.
Regarding Q1: have you tried to find the distributive and annullator morphisms for the putative 2-rig structure on $\widehat{C}$?
I was trying to find at least one distributive morphism, and I have no idea how to reduce $F\hat{\otimes}(H\hat{\oplus} K)$ to/from $F\hat{\otimes} H \,\hat{\oplus}\, F\hat\otimes K$, if $F,H,K : \widehat{C}$. If I'm not wrong (this is very back-of-the-envelope coend calculus),
$$\begin{align*}
F\hat\otimes H &= \int^{UA}FU\times HA\times [\_, U\otimes A]\\
F\hat\otimes K &= \int^{U'B}FU'\times KB\times [\_, U'\otimes B]
\end{align*}$$
whereas
$$\begin{align*}
F \hat\otimes \,(H\hat\oplus K) &= \int^{UV} FU \times (H\hat\oplus K)V \times [\_, U\otimes V] \\ 
&=\int^{UVAB} FU \times HA \times KB \times [V, A\oplus B] \times [\_, U\otimes V] \\ 
&=\int^{UAB} FU \times HA \times KB \times [\_, U\otimes (A\oplus B)] \\ 
&=\int^{UAB} FU \times HA \times KB \times [\_, U\otimes A \oplus U\otimes B] \\ 
\end{align*}$$
...and now we're stuck, unless we have either

*

*R1. a compatibility between $\oplus$ and $\times$, perhaps another distributive morphism;

*R2. a siftedness condition ensuring that
$$ \int^{UA}FU\times HA\times [\_, U\otimes A] \oplus \int^{U'B}FU'\times KB\times [\_, U'\otimes B]$$
can be reduced to an integral on just $U$.

Actually, you need both in order for the computation to proceed; but the conjunction of R1 and R2 is quite strong, as you can see.

Edit: the situation with annullators (for Laplaza, morphisms ${\bf 0}\otimes X \to {\bf 0}$ and $X\otimes {\bf 0} \to \bf 0$) is even worse!
Let's open $F \hat\otimes {\bf 0}$ recalling that in this case $\bf 0$ is the representable $y{\bf 0}$ on the additive unit of $C$:
$$\begin{align*}
\int^{UV} FU \times [V,{\bf 0}] \times [\_,U\otimes V]  
&=\int^U FU \times [\_, U\otimes {\bf 0}] \\ 
&\overset{\rho_U}\to\int^U FU \times [\_, {\bf 0}]\\
&=\varinjlim F \times  [\_, {\bf 0}]
\end{align*}$$
the cartesian structure on $\sf Set$ now entails that this is $\bf 0$ if and only if either factor is empty, but I see no way in which this can be or even map into $y{\bf 0}$ again, as it should.
