= Solution
{c}
First fix a convention that is implicit in the source. For the stated free <generating functional>
$$
Z_0/Z_0[0]=\exp\left(i\bar\eta S\eta+\frac i2J\Delta J\right),
$$
the time-ordered <Wick contractions> are $S_F=\langle T\psi\bar\psi\rangle_0=-iS$ and $\Delta_F=\langle TAA\rangle_0=-i\Delta$. These follow directly by the source differentiation rules in part (i). Thus the kernels called <propagators> in the question are distinguished here from the contractions, which already include their factors of $i$.
At first order in the <QED> interaction, write $a$ for its integration point. The normalized three-point <correlation function> is
$$
G_{\nu\alpha\beta}=-ie\int d^4a\,\Delta_{F\nu\mu}(x-a)\gamma^\mu_{rs}\langle T\{\psi_\alpha(y)\bar\psi_\beta(z)\bar\psi_r(a)\psi_s(a)\}\rangle_0+O(e^3).
$$
The zeroth-order term vanishes because the free <photon> one-point <correlation function> is zero. There are two nonzero <Wick contractions> of the four fields:
$$
\langle T\{\psi_\alpha(y)\bar\psi_\beta(z)\bar\psi_r(a)\psi_s(a)\}\rangle_0
=-S_{F\alpha\beta}(y-z)S_{Fsr}(0)+S_{F\alpha r}(y-a)S_{Fs\beta}(a-z).
$$
The first term pairs the two external <fermions> directly and closes the two fields at the interaction into a <fermion loop>; its minus sign is the <closed fermion loop sign>. The second term joins the external <fermions> through the interaction, giving the connected contribution. This is also obtained by differentiating the quartic term $(i\bar\eta S\eta)^2/2$ in the free <generating functional>; the two choices of bilinear cancel its factor $1/2$.
Insert $S_F=-iS$ and $\Delta_F=-i\Delta$. Since $(-ie)(-i)^3=e$, the result is exactly
$$
\boxed{G_{\nu\alpha\beta}=-e\int d^4a\,\Delta_{\mu\nu}(a-x)\left(\operatorname{tr}[\gamma^\mu S(0)]S_{\alpha\beta}(y-z)-[S(y-a)\gamma^\mu S(a-z)]_{\alpha\beta}\right)+O(e^3).}
$$
The trace term is a disconnected product of a photon <tadpole diagram> with the free external <fermion> <propagator>; the other term is the connected three-point <correlation function>. In this <QED> calculation the disconnected term vanishes, as established next. With a convention using $S_F,\Delta_F$ themselves as the symbols for the <propagators>, the equivalent formula retains the prefactor $-ie$ and the connected-minus-trace ordering above; mixing these two conventions would give an erroneous overall phase.
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