About your second question. If I understand things correctly, you want to solve the Dirichlet problem $$ \Delta u = \text{div} F, \quad u_{\vert \partial \Omega}=0. $$$$ \Delta u = \operatorname{div} F, \quad u_{\vert \partial \Omega}=0. $$ Since your open set is smooth, the parametrix for this problem is a classical pseudo-differential operator with order $-2$ on the the manifold $\Omega$ (with boundary $\partial \Omega$) and that operator has $L^p$ continuity properties for $p\in (1,+\infty).$ You could say in particular that since $\text{div} F$$\operatorname{div} F$ belongs to $W^{-1, p}(\Omega)$, you can solve the Dirichlet problem and get a solution in $W^{1, p}_0(\Omega)$.
For your first question, there is a standard difficulty with the space $L^1$ which is poorly behaved with respect to singular integrals. For instance the Hilbert transform, i.e. the convolution with $\text{pv}(i/π x)$$\operatorname{pv}(i/π x)$, or the Fourier multiplier $\text{sign}\xi$$\operatorname{sign}\xi$ is not bounded on $L^1(\mathbb R)$. Take for instance a function $u$ in $L^1$ with integral 1. The Fourier transform $\hat u$ is a continuous function such that $\hat u(0)=1$. Now consider $$ \phi(\xi)=\text{sign}\xi\times \hat u(\xi). $$$$ \phi(\xi)=\operatorname{sign}\xi\times \hat u(\xi). $$ The inverse Fourier transform of $\phi$ is the Hilbert transform of $u$ and does not belong to $L^1$, otherwise it would be a continuous function, which is not the case since $$\phi(0_+)=\hat u(0)=1,\quad \phi(0_-)=-\hat u(0)=-1.$$
