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 answer to Question (ii) is positive. That is to say, there is a weak equivalences between the following functors from Segal topological categories to quasicategories: the composition of the singular complex functor with the homotopy coherent nerve functor, and the realization of the singular complex.
Indeed, the first step in both functors is the same: we take the singular complex of a Segal topological category, which yields a Segal category.
Therefore, the problem reduces to establishing a weak equivalence between the homotopy coherent nerve functor N and the realization functor R,
both considered as functors from Segal categories to quasicategories.
Both functors are homotopy cocontinuous: the homotopy coherent nerve is a Quillen equivalence, and the realization functor by construction.
Since the quasicategory of quasicategories is a reflective localization of ∞-presheaves on Δ, it suffices to construct a weak equivalence between the restructions of N and R along the Yoneda embedding of Δ into Segal categories.
Indeed, both restrictions are weakly equivalent to the Yoneda embedding of Δ into quasicategories, by construction.
Question (i) is not formulated rigorously, but there are rigorously defined functors from quasicategories to Segal categories. For example, one can use the left adjoint of the homotopy coherent nerve functor, which tautologically provides a positive answer.
Other constructions can be obtained by passing from quasicategories to Segal spaces, and then to Segal categories using the constructions of Joyal–Tierney and Bergner. Bergner's book has details of these construction. To see that the resulting functor is indeed the inverse of the functors considered above, consider the two compositions, and show they are weakly equivalent to the corresponding identity functor by restricting along the Yoneda embedding and observing that the resulting restrictions are weakly equivalent by construction.
Best Answer
Yes, additive categories are $Ab$-enriched categories with all finite direct sums. $Ab$-enriched categories have all 2-limits and 2-colimits by enriched category theory and in particular all bilimits and bicolimits. Additive categories are bireflective inside $Ab$-enriched categories, so they have bilimits and bicolimits as well.