= Solution
Using $\mu^2=\lambda v^2$, expand the potential around the chosen <vacuum>:
$$
V=V_{\min}+\lambda v^2h^2+\lambda vh(h^2+\pi^2)
+\frac\lambda4(h^2+\pi^2)^2.
$$
The linear term vanishes at the minimum. With canonical real-field kinetic terms, comparison with $m_h^2h^2/2+m_\pi^2\pi^2/2$ gives
$$
\boxed{m_h^2=2\lambda v^2=2\mu^2,\qquad m_\pi^2=0.}
$$
The massive radial scalar changes the magnitude of the <vacuum> field; the massless <Goldstone boson> changes its orientation. The symmetry is global here, so no gauge field eats that mode.
The genuinely new scalar cubic interactions are $-\lambda v h^3-\lambda v h\pi^2$. Including identical-field factorials, their <Feynman vertex> factors are
$$
\boxed{hhh:\ -6i\lambda v,\qquad h\pi\pi:\ -2i\lambda v.}
$$
The quartic interactions remain $-\lambda h^4/4-\lambda h^2\pi^2/2-\lambda\pi^4/4$, with respective vertex factors $-6i\lambda,-2i\lambda,-6i\lambda$. The shifted Yukawa terms from <3/a/iv> give $h\bar\psi\psi$ vertex $-ig$ and $\pi\bar\psi\psi$ vertex $g\gamma_5$. These Yukawa structures were present before the shift; the new <fermion> bilinear is the <mass term>, not an additional independent interaction coupling. There are no derivative interactions from the linear Cartesian field shift.
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