Examples on product of weakly differentiable functions 
*

*Are there two functions $f,g \in L^1_\text{loc}(\mathbb{R}^N)$, $N\ge 1$, such that $f,g$ are weakly differentiable (have all first-order weak partial derivatives in $L^1_\text{loc}(\mathbb{R}^N)$) but $fg$ has no weak partial derivative?

*Are there two functions $f,g \in L^1_\text{loc}(\mathbb{R}^N)$, $N\ge 1$, such that $f,g$ are weakly differentiable (have all first-order weak partial derivatives in $L^1_\text{loc}(\mathbb{R}^N)$) but $fg \in L^1_\text{loc}(\mathbb{R}^N)$ and has no weak partial derivative?

A good place to look for examples is the power functions. This is because the local integrability of these functions (or lack thereof) is easily determined by comparing the power to the dimension. To address your first item, use the fact that $f(x) = |x|^{-p}$ is locally integrable if and only if $p< N$. For such $f$ we have 
\begin{equation*}
|\nabla f(x)| = p|x|^{-(p + 1)}
\end{equation*}
and consequently, (choosing $g = f$), 
\begin{equation*}
|\nabla (f(x)g(x))| = 2p|x|^{-(2p + 1)}. 
\end{equation*}
Thus, both $\nabla f$ and $\nabla g$ are locally integrable as long as $p + 1< N$, and $fg$ fails to be locally integrable as long as $2p + 1\geq N$. Now choose any $p$ such that $(N - 1)/2\leq p< N-1$. This argument works for $N>1$.
