= Solution
{c}
Choose $D_\mu=\partial_\mu-igA_\mu^at^a$ on the <Dirac field>, $t^a=\sigma^a/2$ in the <fundamental representation> of <SU(2)>, and $[t^a,t^b]=i\epsilon^{abc}t^c$. Define the <gauge field strength> and the <Adjoint representation> <gauge covariant derivative> by
$$
F_{\mu\nu}^a=\partial_\mu A_\nu^a-\partial_\nu A_\mu^a
+g\epsilon^{abc}A_\mu^bA_\nu^c,\qquad
(D_\mu c)^a=\partial_\mu c^a+g\epsilon^{abc}A_\mu^bc^c.
$$
The required <gauge-fixed Yang-Mills Lagrangian in a covariant gauge>, including the matter term, is
$$
\boxed{\mathcal L=-\frac14F_{\mu\nu}^aF^{a\mu\nu}
+\bar\psi(i\gamma^\mu D_\mu-m)\psi
-\frac1{2\xi}(\partial^\mu A_\mu^a)^2
-\bar c^a\partial^\mu(D_\mu c)^a.}
$$
Here $c^a$ and $\bar c^a$ are independent <Grassmann fields> in the <Adjoint representation>: the <Faddeev-Popov ghost fields> and <antighost fields>. Under an infinitesimal <gauge transformation>, $\delta A_\mu^a=(D_\mu\omega)^a$; hence the <Lorenz gauge> function $G^a=\partial^\mu A_\mu^a$ varies by $\partial^\mu D_\mu^{ab}\omega^b$. Its <Faddeev-Popov determinant> is represented by the ghost term, with an irrelevant field-independent determinant sign absorbed into normalization. <Integration by parts> gives $\mathcal L_{\rm gh}=(\partial^\mu\bar c^a)(D_\mu c)^a$, explicitly showing the <ghost-gluon vertex>.
There is no scalar vacuum expectation value here, so <R-xi gauge> reduces to the linear <covariant gauge> displayed above, with no <Goldstone boson> term. At $\xi=1$ the <Feynman gauge> <gauge boson> <propagator> is $-i\delta^{ab}g_{\mu\nu}/(k^2+i0)$. With Fourier convention $e^{-ikx}$ and all three boson <four-momenta> incoming, this choice gives exactly the source's $+g\epsilon^{abc}$ three-boson <Feynman rule>, and the matter <interaction vertex> is $+ig\gamma^\mu t^a$. The sign of $D_\mu$ has been explicitly chosen here; the QED convention in question 2 instead has vertex $-ie\gamma^\mu$.
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