= Solution
For the <ghost coupling in a linear Cartesian Abelian Higgs gauge>, the common factor in the <Scalar field>[scalar] parametrization cancels from the <gauge transformation>. Separating real and imaginary parts gives
$$
\delta h=g\theta b,\qquad \delta b=-g\theta(v+h),\qquad \delta(\partial\cdot A)=-\Box\theta.
$$
Consequently
$$
\mathcal M=-\Box+\xi g^2v(v+h),\qquad
\boxed{\mathcal L_{\mathrm{gh}}=\bar c\Box c-\xi g^2v^2\bar cc-\xi g^2v h\bar cc.}
$$
For the variables and <Gauge fixing>[gauge function] actually printed, the <Faddeev-Popov ghost>[ghost mass] and <ghost coupling in a linear Cartesian Abelian Higgs gauge>[Higgs-ghost] <vertex> are therefore
$$
\boxed{m_c^2=\xi g^2v^2,\qquad V_{h\bar cc}=-i\xi g^2v.}
$$
For the usual nonnegative <Gauge-fixing parameter>[gauge parameter], $m_c=\sqrt\xi\,g|v|$. The momentum-space <ghost propagator> is $-i/(k^2+m_c^2-i0)$ in this kinetic convention. There is no direct <Faddeev-Popov ghost>[ghost-photon] <vertex> or ghost-$b$ <vertex>. At $\xi=0$ the <Faddeev-Popov ghost>[ghost] becomes massless and decouples.
\b[The <Scalar field>[scalar] prefactor $1/4$ is incompatible with the stated <Canonical field normalization>[canonical] broken-vacuum interpretation.] At $h=b=0$ it gives $|\phi|^2=v^2/16$, whereas the <Scalar potential>[potential] is minimized at $|\phi|^2=v^2/2$. In fact the printed parametrization gives
$$
\left.\frac{\partial V}{\partial h}\right|_{h=b=0}=-\frac{7\lambda v^3}{256}\ne0.
$$
Its $h,b$ <kinetic terms> have coefficient $-1/16$, not $-1/2$, and its vector <Mass>[mass parameter] is $g^2v^2/8$. The given <Gauge fixing>[gauge function] consequently does not cancel the vector-scalar mixing in those literal variables. The <Faddeev-Popov ghost>[ghost] operator computed above nevertheless follows algebraically from that <Gauge fixing>[gauge function] without changing the printed fields.
With a <canonical Abelian Higgs shift> $\phi=(v+h+ib)/\sqrt2$, the expansion is about a true <minimum>, the vector <mass> is $m_A=g|v|$, the <gauge fixing> cancels the mixing, and the same <Faddeev-Popov ghost>[ghost] formulas acquire the usual interpretation $m_c^2=\xi m_A^2$. For the literal $1/4$ parametrization, a <Canonical field normalization>[canonical] fluctuation is $h_{\mathrm{can}}=h/\sqrt8$, so its <Faddeev-Popov ghost>[ghost] <vertex> would instead be $-i\sqrt8\,\xi g^2v$; the non-vacuum background problem still remains. These two interpretations must not be silently mixed.
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