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.
The one line answer is that the category $\mathsf{Ab}$ of abelian groups is enriched over the skew-monoidal category $\mathsf{Gp}$ of groups, and that this "faux-tensor" defines a skew-action of the skew-monoidal category $\mathsf{Gp}$ on $\mathsf{Ab}$.
A skew-monoidal structure on a category $\mathcal{C}$ consists of a "tensor product" functor $\boxtimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$, a "unit" object $I \in \mathcal{C}$, and "associativity and unit constraint" natural transformations $\alpha \colon (X \boxtimes Y) \boxtimes Z \to X \boxtimes (Y \boxtimes Z)$, $\lambda \colon I \boxtimes X \to X$, and $\rho \colon X \to X \boxtimes I$, satisfying the original five coherence axioms of Mac Lane. The important point is that these associativity and unit constraints are not required to be invertible. This notion was introduced by Szlachányi in his paper
Kornél Szlachányi. Skew-monoidal categories and bialgebroids. Adv. Math. 231 (2012), no. 3-4, 1694--1730. https://doi.org/10.1016/j.aim.2012.06.027
and has been much studied since, especially by the Australian school of category theory.
The "half-tensor products" of groups that you describe are part of a skew-monoidal structure on the category $\mathsf{Gp}$ of groups. This skew-monoidal structure is an instance of the family of examples described in Example 2.7 of my paper:
Alexander Campbell. Skew-enriched categories. Applied Categorical Structures 26 (2018), no. 3, 597--615. https://doi.org/10.1007/s10485-017-9504-0
The tensor product $G \boxtimes H$ of two groups $G$ and $H$ is the group you denote by $G \triangleleft H$, i.e. the copower of $G$ by the underlying set of $H$. Note that group homomorphisms $G \boxtimes H \to K$ correspond to functions $G \times H \to K$ that are group homomorphisms in the first variable. The unit object is the free group on one generator, i.e. $\mathbb{Z}$. The associativity and unit constraints are a little more complicated to describe, but suffice it to say that they are not invertible.
This skew-monoidal structure on $\mathsf{Gp}$ is closed: the functor $- \boxtimes H$ has a right adjoint which sends a group $K$ to the group $[H,K]$ of all functions from $H$ to $K$ with the pointwise group structure; this group $[H,K]$ is the internal hom for this skew-monoidal structure on $\mathsf{Gp}$. Thus $\mathsf{Gp}$ is also a skew-closed category in the sense introduced by Ross Street in his paper:
Ross Street. Skew-closed categories. J. Pure Appl. Algebra 217 (2013), no. 6, 973--988. https://doi.org/10.1016/j.jpaa.2012.09.020
Now, just as one can define categories enriched over monoidal categories, one can also define categories enriched over skew-monoidal categories. (In the terminology of my paper cited above, this is the same thing as a "left normal skew-enrichment" over the skew-monoidal category. Enrichment over skew-closed categories is defined in Street's paper cited above.)
We can define an enrichment of $\mathsf{Ab}$ over the above skew-monoidal structure on $\mathsf{Gp}$ as the change of base of the usual self-enrichment of $\mathsf{Ab}$ along the inclusion functor $\mathsf{Ab} \to \mathsf{Gp}$ equipped with the lax monoidal structure whose tensor constraint $A \boxtimes B \to A \otimes B$ is the homomorphism $U(B) \odot A \to A \otimes B$ whose component at an element $b \in B$ is $-\otimes b \colon A \to A \otimes B$.
Unpacking this, we have that, for each pair of abelian groups $A$ and $B$, the hom-group $\underline{\operatorname{Hom}}(A,B)$ is the usual group of group homomorphisms from $A$ to $B$, with its pointwise group structure, but where we have forgotten that it's abelian. For each triple of abelian groups $A$, $B$, and $C$, the composition homomorphism $\underline{\operatorname{Hom}}(B,C) \boxtimes \underline{\operatorname{Hom}}(A,B) \to \underline{\operatorname{Hom}}(A,C)$ corresponds to the usual composition function $\operatorname{Hom}(B,C) \times \operatorname{Hom}(A,B) \to \operatorname{Hom}(A,C)$, but where we have forgetten that it's a group homomorphism in the second variable. Similarly, the unit homomorphisms $\mathbb{Z} \to \underline{\operatorname{Hom}}(A,A)$ simply pick out the identity homomomorphisms.
(Note that this enrichment of $\mathsf{Ab}$ over $\mathsf{Gp}$ can also be seen an instance of Example 2.7 of my paper cited above, since the category of abelian groups is equivalent to the category of group objects in $\mathsf{Gp}$.)
As you've spelled out in your question, the hom-functor $\underline{\operatorname{Hom}} \colon \mathsf{Ab}^\mathrm{op} \times \mathsf{Ab} \to \mathsf{Gp}$ is part of a two-variable adjunction, and so there are defined tensoring and cotensoring operations of an abelian group by a group. In particular, the tensoring operation defines a skew-action of the skew-monoidal category $\mathsf{Gp}$ on the category $\mathsf{Ab}$, in the sense of the paper:
Stephen Lack and Ross Street. Skew-monoidal reflection and lifting theorems. Theory Appl. Categ. 30 (2015), Paper No. 28, 985--1000. http://tac.mta.ca/tac/volumes/30/28/30-28abs.html
Note that a skew-action of a skew-monoidal category $\mathcal{V}$ on a category $\mathcal{C}$ is simply an oplax monoidal functor $\mathcal{V} \to \operatorname{Fun}(\mathcal{C},\mathcal{C})$.
Best Answer
$\newcommand{\defeq}{\overset{\mathrm{def}}{=}}\newcommand{\id}{\mathrm{id}}\newcommand{\Mod}{\mathrm{Mod}}\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}\newcommand{\d}{\mathrm{d}}\newcommand{\dAb}{\mathsf{End}(\mathsf{Ab})}$DGAs are monoids in chain complexes. To get differential rings as monoids in some monoidal category, it suffices to remove the grading and the $\d^{2}=0$ condition.
In detail, consider the category $\mathsf{End}(\mathsf{Ab})\defeq\mathsf{Fun}(\mathbf{B}\N,\mathsf{Ab})$ whose
We can then put a monoidal structure $\otimes_\Z$ on $\dAb$ by defining $$(A,\d_A)\otimes(B,\d_B)=(A\otimes_\Z B,\d_A\otimes_\Z1_B+1_A\otimes_\Z\d_B),$$ where the unit is given by the pair $(\Z,\d_\Z)$ with $\d_\Z\overset{\mathrm{def}}{=} 0$. Note that a morphism in $\dAb$ from $(\Z,\d_\Z)$ to $(A,\d_A)$ is just a "constant" element of $A$, i.e. an element with $\d_A a = 0$.
A monoid in $(\dAb,\otimes_\Z,(\Z,\d_\Z))$ will then be a triple $((A,\d),\mu,\eta)$ with
such that the usual associativity and unitality diagrams commute, which makes $(A,\mu,\eta)$ into a ring, and together with $\d$, this makes the quadruple $((A,\d),\mu,\eta)$ into a differential ring.