= Solution
Split the <string embedding map> into its constant <worldsheet zero mode> and orthogonal fluctuations, $X^\mu=x^\mu+X'^\mu$. The kinetic operator has no inverse on the constant mode, while its inverse on $X'$ is the <worldsheet Green function> $G$. Completing the square in the <Gaussian functional integral> gives
$$
\int\mathcal DX'\,e^{-S[X']-\int J\cdot X'}
\propto\exp\left(\frac12\int_{\Sigma\times\Sigma}d^2z\,d^2w\,J(z)G(z,w)J(w)\right).
$$
The remaining ordinary integral is
$$
\int d^Dx\,e^{-x^\mu\int_\Sigma d^2zJ_\mu(z)},
$$
which proves (1), up to a source-independent <functional determinant>.
To insert $n$ <tachyon vertex operators>, choose
$$
J_\mu(z,\bar z)=-i\sum_{j=1}^nk_{j\mu}\delta^{(2)}(z-z_j).
$$
The zero-mode integral produces target-space momentum conservation,
$$
(2\pi)^D\delta^{(D)}\left(\sum_jk_j\right).
$$
In the nonzero-mode integral, discard the coincident self-contractions by <normal ordering>. With $G(z,w)=-(\alpha'/2)\log|z-w|^2$, the remaining pairwise contractions produce the <Koba-Nielsen factor>, so
$$
\boxed{
\left\langle\prod_{j=1}^ne^{ik_j\cdot X(z_j,\bar z_j)}\right\rangle
\propto(2\pi)^D\delta^{(D)}\left(\sum_jk_j\right)
\prod_{i<j}|z_i-z_j|^{\alpha'k_i\cdot k_j}
}.
$$
For three closed-string tachyons on the sphere, $k_i^2=4/\alpha'$ and $\sum_i k_i=0$, hence $\alpha'k_i\cdot k_j=-2$ for $i\ne j$. The matter correlator is therefore $|z_{12}z_{13}z_{23}|^{-2}$. Gauge fixing the sphere's <Möbius transformation> group fixes three insertion points and supplies the <bc ghost system> correlator $|z_{12}z_{13}z_{23}|^2$, exactly cancelling this position dependence. Thus
$$
\boxed{\mathcal A_3=C_{S^2}g_c^3(2\pi)^D\delta^{(D)}(k_1+k_2+k_3)}.
$$
Here each vertex contributes one closed-string coupling $g_c$, $C_{S^2}$ contains the sphere vacuum normalization and conventions, the <worldsheet zero mode> gives the momentum delta function, the nonzero modes give the <Koba-Nielsen factor>, and the ghost determinant divides by the conformal Killing group. Since $C_{S^2}\propto g_s^{-2}$ and $g_c\propto g_s$, the net <string genus expansion> dependence is $g_s$.
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