= Solution
Expand the <gauge covariant derivative> in the scalar kinetic term:
$$
(D_\mu\varphi)^*D^\mu\varphi
=\partial_\mu\varphi^*\partial^\mu\varphi
+ieA_\mu\bigl(\varphi\partial^\mu\varphi^*-\varphi^*\partial^\mu\varphi\bigr)
+e^2A_\mu A^\mu\varphi^*\varphi.
$$
The cubic interaction is therefore
$$
\boxed{\mathcal L_3=ieA_\mu(\varphi\partial^\mu\varphi^*-\varphi^*\partial^\mu\varphi).}
$$
For an incoming scalar momentum $p$ and outgoing momentum $q$, derivatives give $-ip$ on the incoming field and $+iq$ on its outgoing conjugate. Multiplication by the factor $i$ from the interaction action gives the <scalar electrodynamics three-point vertex> $-ie(p+q)_\mu$.
The other vertex comes from
$$
\boxed{\mathcal L_4=e^2A_\mu A^\mu\varphi^*\varphi.}
$$
Differentiating twice with respect to the identical photon fields gives a factor of two, so the <seagull vertex> is $+2ie^2\eta_{\mu\nu}$. There is no additional factor of two from the scalar fields, since the <complex scalar field> and its conjugate are distinct legs.
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