Let $u$, $\omega\in L^1_{\text{loc}}(\mathbb R^N$) be given. We further assume that $\omega\geq 1$, l.s.c, and satisfies, for a constant $C>0$, $$ \frac{1}{|B(x,r)|}\int_{B(x,r)}\omega(y)dy\leq C\omega(x) $$ for any $x\in\mathbb R^N$ and $r>0$.
My question 1: do we have $u\omega\in L^1_{\text{loc}}(\mathbb R^N)$ as well? I feel not but can not find an counterexample...
Since you put no assumptions on $u$ apart from $L_{\rm loc}^1$, the question could be recast as: what functions $\omega$ are multipliers on $L_{\rm loc}^1$, in the sense that $u\in L_{\rm loc}^1\implies u\omega \in L_{\rm loc}^1$?
The answer to which is: precisely the locally bounded functions, $L_{\rm loc}^\infty$. The argument is essentially the same as for showing that the multiplier space of $L^1$ is $L^\infty$, which I recast for convenience. Given a function $\omega$ that is not essentially bounded, let $ u = \sum c_n \chi_{\{\omega> n\}}$ where $c_n$ are chosen so that the series just barely converges in $L^1$ norm, so that $\sum n c_n \chi_{\{\omega> n\}}$ diverges in $L^1$.
Examples of functions that satisfy your condition but are not locally bounded were given in comments: e.g., Ian suggested $|x|^{-1/2}$.