Your question requires some clarifications to be answered precisely. Your first statement: I know that the existence of a solution to the above Dirichlet problem depends both on the regularity of $\partial U$ and on the choice of $g$ seems to imply that you are considering a general open set $U$.
In that case, you should explain what the identity $u=g$ on $\partial \Omega$ means (in the sense of traces for regular boundaries, but in general?).
Next, you define a Green function by introducing $\phi^x$, and further down, you talk of a normal derivative of $G$, $\partial_n G$ : should we therefore understand that in fact the domain $U$ has a regular boundary, where a normal exists ?
If you are happy with a regular boundary (e.g. satisfying an exterior cone condition in dimension $\geq3$, and less in dimension $2$, see the related question here), the classical Dirichlet problem is uniquely solvable for any $g\in C^0(\bar{U})$, see Gilbarg & Trudinger Chapter 2 for example. Then, provided that the Green formula you wrote makes sense (a normal exists etc.) you have constructed a Green function as well.