= Solution
Choose a nonconstant element $t$ of the <function field> $k(C)$. By Question 1 it gives a <finite morphism> $\pi:C\to\mathbb P^1$. The inverse images $U_0=\pi^{-1}(\mathbb A^1_t)$ and $U_1=\pi^{-1}(\mathbb A^1_{t^{-1}})$ are affine, because the inverse image of an affine under a <finite morphism> is the spectrum of a finite algebra over its coordinate ring. They cover $C$. This proves the <two-affine cover of a smooth projective curve> assertion.
Their intersection is affine by Question 2. Thus the <Čech cochain complex> for any <coherent sheaf> $\mathcal F$ has just two nonzero terms,
$$
\Gamma(U_0,\mathcal F)\oplus\Gamma(U_1,\mathcal F)
\longrightarrow\Gamma(U_0\cap U_1,\mathcal F).
$$
It follows that
$$
\boxed{H^i(C,\mathcal F)=0\quad(i\ge2).}
$$
On the <projective line>, put $t=X_1/X_0$ on the first chart. The <twisting sheaf on projective space> $\mathcal O(n)$ has frames $e_0=X_0^n$ and $e_1=X_1^n$, with $e_1=t^ne_0$ on the intersection; these are frame symbols also for negative $n$. In the $e_0$ frame the differential of the <Čech cochain complex> is
$$
k[t]\oplus k[t^{-1}]\longrightarrow k[t,t^{-1}],\qquad
(a,b)\longmapsto t^nb-a.
$$
A <global section> is therefore a Laurent polynomial in both $k[t]$ and $t^nk[t^{-1}]$. Its monomials have exponents $0\le j\le n$. For $n\ge0$ the sections have basis $e_0,te_0,\ldots,t^ne_0$; for $n<0$ there are none. The first <sheaf cohomology> is
$$
H^1(\mathbb P^1,\mathcal O(n))=
\frac{k[t,t^{-1}]}{k[t]+t^nk[t^{-1}]}.
$$
Its basis consists of the monomials $t^je_0$ with $n<j<0$. These are precisely the exponents missed by the two chart spaces. Thus for every integer $n$,
$$
\boxed{h^0(\mathbb P^1,\mathcal O(n))=\max(n+1,0),\qquad
h^1(\mathbb P^1,\mathcal O(n))=\max(-n-1,0).}
$$
Both groups vanish at $n=-1$; all higher groups vanish for every $n$. This is also an explicit calculation of the <Čech cohomology of twists on the projective line>.
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