= Solution
Use the <Minkowski metric>, path-integral weight $e^{iS}$ and the <Abelian gauge theory> transformation $A_\mu\mapsto A_\mu+\partial_\mu\omega$. The gauge functional $F[A]=\partial\cdot A+A^2$ varies as
$$
\delta_\omega F=(\Box+2A^\mu\partial_\mu)\omega.
$$
Thus the <Faddeev-Popov operator> is $\mathcal M_A=\Box+2A\cdot\partial$. It depends on the gauge field despite the gauge group being Abelian: \b[the ghosts interact because this gauge condition is nonlinear].
Choose the <gauge-fixing fermion> $\Psi=\int d^4x\,\bar c(F[A]+\xi h/2)$. The <gauge-fixed action> $S=S_{\rm Maxwell}+s\Psi$ is
$$
\boxed{S=\int d^4x\left[-\frac14F_{\mu\nu}F^{\mu\nu}
+h(\partial\cdot A+A^2)+\frac\xi2h^2
-\bar c(\Box+2A\cdot\partial)c\right].}
$$
Here the tensor $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ is distinct from the scalar gauge functional $F[A]$. With $\xi=0$, integrating over $h$ imposes the exact printed constraint. For nonzero $\xi$, eliminating $h=-F[A]/\xi$ instead gives
$$
\mathcal L=-\frac14F_{\mu\nu}F^{\mu\nu}-\frac1{2\xi}(\partial\cdot A+A^2)^2
-\bar c\Box c-2\bar c A^\mu\partial_\mu c.
$$
This version displays the additional gauge-dependent cubic and quartic gauge-field vertices as well as the ghost interaction; the strict condition is its $\xi\to0$ limit.
For the <Fourier transform> convention $c(x)=\int d^4p\,(2\pi)^{-4}e^{-ipx}c(p)$, the quadratic ghost kernel is $p^2$. With the ordering $\langle c(p)\bar c(q)\rangle$,
$$
\boxed{\langle c(p)\bar c(q)\rangle=(2\pi)^4\delta^{(4)}(p+q)\frac{i}{p^2+i0}.}
$$
The term $-2\bar c A^\mu\partial_\mu c$ has Fourier coefficient $2ip_\mu$, where $p$ is the incoming ghost momentum. Multiplication by $i$ in the <Feynman rule> gives
$$
\boxed{V_\mu(\bar c(q),c(p),A(k))=-2p_\mu,\qquad p+q+k=0.}
$$
These signs refer to the displayed action, Fourier convention and ghost ordering. Reversing the ghost/antighost convention changes corresponding signs consistently. A closed <ghost loop> has the additional minus sign from <Grassmann variables>.
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