= Solution
A <meromorphic differential on a Riemann surface> has local expression $\omega=a(z)dz$ with a meromorphic coefficient. On overlapping coordinates it transforms by $a(z)dz=\widetilde a(w)dw$. For a <meromorphic function> $f$, the coefficients $f'(z)$ obey this transformation by the chain rule, so $df$ is such a differential. Two nonzero differentials have coefficient ratio $a/b$; the derivative factors cancel on overlaps, yielding a global <meromorphic function> $h$ with $\eta=h\omega$.
An invertible holomorphic coordinate change has a nonzero holomorphic derivative, so multiplying by its derivative changes no local zero or <pole> order. Thus the differential's divisor is well defined. For a compact connected surface, the canonical-degree theorem states $\sum_P\operatorname{ord}_P\omega=2g-2$, including negative orders for <poles>. Therefore $\boxed{g=1+\tfrac12\sum_P\operatorname{ord}_P\omega}$.
The projective closure has equation $U^2-V^2-W^2=0$. Its gradient $(2U,-2V,-2W)$ cannot vanish at a projective point, so it is nonsingular. At infinity $W=0$ gives the two points $[1:1:0]$ and $[1:-1:0]$. In the supplied sphere parameter,
$$
v(t)=\frac{2t}{t^2-1},\qquad dv=-\frac{2(t^2+1)}{(t^2-1)^2}\,dt.
$$
This has double <poles> at $t=\pm1$, the two infinity points, and simple zeros at $t=\pm i$. At the parameter infinity use $s=1/t$, giving $v=2s/(1-s^2)$ and $dv=2(1+s^2)(1-s^2)^{-2}ds$, which is regular and nonzero at $s=0$. Hence \b[$dv$ extends meromorphically over the whole projective curve]. Its order sum $2-4=-2$ gives genus zero, consistently with the stated biholomorphism from the sphere.
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