= Solution
The Euclidean <dilaton> term is $S_\Phi=(4\pi)^{-1}\int\sqrt h\,\Phi(X)R^{(2)}$. A constant mode $\Phi_0$ therefore gives $S_\Phi=\Phi_0\chi$ by the <Gauss-Bonnet theorem>. Setting the <string coupling> to $g_s=e^{\Phi_0}$, an oriented closed worldsheet of genus $g$ has weight
$$
e^{-S_\Phi}=g_s^{-\chi}=g_s^{2g-2}.
$$
Each canonically normalized external closed-string vertex supplies a further factor $g_s$. Thus the connected $n$-point genus-$g$ contribution carries
$$
\boxed{g_s^{2g-2+n}.}
$$
This is the <string genus expansion>. Four external strings on a sphere give $g_s^2$, as in the displayed tree amplitude.
Use the conventional complex-coordinate measure $d^2z=2\,d(\operatorname{Re}z)\,d(\operatorname{Im}z)$, with $\partial=(\partial_x-i\partial_y)/2$. The action then has the standard real-coordinate normalization $(4\pi\alpha')^{-1}\int(\nabla X)^2d^2x$. In these conventions the <free-boson worldsheet propagator> is
$$
\boxed{\langle X^\mu(z,\bar z)X^\nu(w,\bar w)\rangle
=-\frac{\alpha'}2\delta^{\mu\nu}\log|z-w|^2,}
$$
up to an additive infrared constant. Indeed the quadratic kernel after integration by parts is $-(\pi\alpha')^{-1}\partial\bar\partial$, and $\partial\bar\partial\log|z-w|^2=2\pi\delta^{(2)}(z-w)$ for this measure. Their product gives the required delta function. A simultaneous change of the coordinate-measure convention and action coefficient leaves this physical normalization unchanged.
Separate the constant mode $x_0$ of $X$. Its integral supplies $(2\pi)^D\delta^{(D)}(\sum_i p_i)$, so <four-momentum conservation> is essential, even when that delta function is suppressed in the amplitude notation. Treat the exponential vertices as normal ordered: self-contractions are then absent. The <Gaussian functional integral> and <Wick theorem> give
$$
\left\langle\prod_{i=1}^4:e^{ip_i\cdot X(z_i)}:\right\rangle
\propto\delta^{(D)}\!\left(\sum_i p_i\right)
\exp\!\left(-\sum_{j<l}p_j\cdot p_l\,\langle X(z_j)X(z_l)\rangle\right)
=\delta^{(D)}\!\left(\sum_i p_i\right)\prod_{j<l}|z_j-z_l|^{\alpha'p_j\cdot p_l}.
$$
The oscillator determinant and normalization constants do not depend on the insertion positions. Substitution therefore gives the requested <Koba-Nielsen factor> under the four insertion integrals, with the sphere's residual conformal-group volume divided out.
For the Möbius transformation $z'=(az+b)/(cz+d)$, the unit determinant condition gives
$$
z'_j-z'_l=\frac{z_j-z_l}{(cz_j+d)(cz_l+d)},\qquad
d^2z'_i=|cz_i+d|^{-4}d^2z_i.
$$
The distance-product factor attached to insertion $i$ is $|cz_i+d|^{-\alpha'\sum_{j\ne i}p_i\cdot p_j}$. Momentum conservation changes its exponent to $\alpha'p_i^2$. Thus the total factor at each insertion, including the measure, is
$$
|cz_i+d|^{\alpha'p_i^2-4}.
$$
On the <tachyon> mass shell $p_i^2=4/\alpha'$ it is one. This proves the <Möbius invariance of on-shell closed-string vertex integrals>. \b[The correlator is covariant; the correlator times all four measures is invariant.] This is the invariant integrand needed to quotient by the residual $SL(2,\mathbb C)$ transformations.
For the massless <Type II superstring theory> amplitude, use $s=-(p_1+p_2)^2$ in Lorentzian target signature and $s+t+u=0$. At generic fixed $t$, the numerator factor $\Gamma(-\alpha's/4)$ has simple poles at
$$
\boxed{s=M_n^2=\frac{4n}{\alpha'},\qquad n=0,1,2,\ldots.}
$$
At these points $\Gamma(1+\alpha's/4)=\Gamma(1+n)=n!$ is finite and nonzero. The $t,u$ factors are also finite and nonzero for generic $t$, so the poles persist; isolated residue zeros at special kinematics do not remove the general mass level. Tree-level factorization interprets a pole at $s=M^2$ as exchange of a state of that mass. The <Type II gamma-factor mass poles> therefore exhibit a massless level and an infinite tower of massive levels, with equally spaced squared masses $4/\alpha'$.
There is no negative-$s$ pole in this exchanged tower. In particular, at negative integer values of $\alpha's/4$ the reciprocal s-channel denominator can instead have zeros. This is consistent with the absence of the bosonic <tachyon> in the <Type II superstring mass spectrum>. The factor alone does not determine degeneracies, spins or every possible uncoupled state; external polarizations and the rest of the amplitude control their residues.
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