@Goulifet The function $1/\Gamma$ is entire, but you are right that to get $(\ast)$, you need a modification (I gave it in a new edit for $-1$ and a modification of the last statement).

Yes, indeed. The constant on the right-hand-side of the normal derivative is forced by Green's formula.

Let me put it that way. You must try to prove your theorem on your vector field $X$ (say uniqueness of weak solutions) for the particular case where $DX$ is as above, which is not so easy. After this, you enter a more technical area, in which you are not stricto sensu reduced to that case, but you can argue via an approximation procedure. The key argument for the uniqueness of weak solutions is a commutator estimate between the vector field and a smoothing operator, and this argument leaves plenty of room for approximation.

As written in my answer on Mathoverflow, taking a $𝐵𝑉$ vector field on $\mathbb R^n$ with absolutely continuous divergence amounts essentially to deal with $𝑋=\sum_{1≤𝑗≤n}𝑎_j(𝑥′,𝑥_n)\frac{\partial}{\partial x_j}$ where $𝑥′\in \mathbb R^{n-1}$ and $\frac{\partial a_1}{\partial x_n}$ Radon measure whereas all other entries of $𝐷𝑋$ are absolutely continuous. Although it is not sufficient, it means that a good idea is to start with $𝑎_1(𝑥_1,𝑥_2)\frac{\partial}{\partial x_1}$ in $\mathbb R^2.$

You should be more specific about what you mean by "reduction".

Yes, it is a general fact for a first-order scalar PDE, which is in fact equivalent to a system of ODE. Lipschitz-continuity is a sufficient condition to get a proper definition for the flow $\phi$ above, but it is not necessary: looking at a system of ODE $\dot x=v(x)$, it is enough to know that $v\in W^{1,1}$ with a bounded divergence to get a flow. The latter situation was studied by P.L. Lions \& R. DiPerna.