= Solution
Treating derivatives with respect to the <Grassmann variables> as graded derivatives and using $\{\partial_\alpha,\theta^\beta\}=\delta_\alpha{}^\beta$ gives
$$
\{\mathcal D_\alpha,\overline{\mathcal D}_{\dot\alpha}\}
=-2i\sigma^\mu_{\alpha\dot\alpha}\partial_\mu
=2\sigma^\mu_{\alpha\dot\alpha}\mathcal P_\mu,
\qquad \mathcal P_\mu=-i\partial_\mu.
$$
The terms in $\{\mathcal D_\alpha,\mathcal D_\beta\}$ and $\{\overline{\mathcal D}_{\dot\alpha},\overline{\mathcal D}_{\dot\beta}\}$ cancel pairwise, so both vanish.
A <chiral superfield> obeys $\overline{\mathcal D}_{\dot\alpha}\Phi=0$, while an <antichiral superfield> obeys $\mathcal D_\alpha\Phi^\dagger=0$. Since $\overline{\mathcal D}_{\dot\alpha}y^\mu=0$ for $y^\mu=x^\mu+i\theta\sigma^\mu\bar\theta$, the general chiral expansion is
$$
\Phi(y,\theta)=\phi(y)+\sqrt2\theta\psi(y)+\theta^2F(y).
$$
In the original coordinates this is
$$
\Phi=\phi+\sqrt2\theta\psi+\theta^2F+i\theta\sigma^\mu\bar\theta\,\partial_\mu\phi
-\frac{i}{\sqrt2}\theta^2\partial_\mu\psi\sigma^\mu\bar\theta
-\frac14\theta^2\bar\theta^2\Box\phi,
$$
with signs following the conventions of the question.
A holomorphic function $W(\Phi)$ is chiral. Its highest component transforms into a spacetime divergence, so the <F-term> action
$$
S_W=\int d^4x\,d^2\theta\,W(\Phi)+\mathrm{h.c.}
$$
is supersymmetric even though it integrates over only chiral half of <superspace>.
The <non-renormalization theorem> says that perturbative loop corrections are full-superspace D-terms and cannot generate a new local superpotential. Holomorphy and spurion symmetries therefore preserve
$$
W_{\rm Wilsonian}=\frac12m\Phi^2+\frac13\lambda\Phi^3.
$$
The Kähler potential is renormalized, however. If its kinetic term is $Z\Phi^\dagger\Phi$, canonical normalization $\Phi_c=Z^{1/2}\Phi$ gives
$$
m_{\rm phys}=\frac mZ,
\qquad
\lambda_{\rm phys}=\frac\lambda{Z^{3/2}},
$$
up to scheme and scale conventions. Thus superpotential parameters are holomorphic invariants while physical masses and couplings still run through <wave-function renormalization>.
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