= Solution
\b[The printed vanishing claim needs the restriction $i\ge1$.] On $C=\mathbb P^1$ with $\mathcal F=\mathcal O$ and $D=n\{\infty\}$, Question 3 gives $h^0(\mathcal F(D))=n+1$ for every $n\ge0$. Thus no bound on the <degree of a divisor> can force vanishing in degree zero. We prove finiteness in every degree and the corrected vanishing in every positive degree, uniformly over all <divisors on an algebraic curve> of large degree.
First take a <finite morphism> $\pi:C\to\mathbb P^1$ as in Question 3 and put $E=\pi_*\mathcal F$. The <direct image sheaf> $E$ is coherent: on an affine chart its module is finitely generated over a finite algebra and hence over the chart's coordinate ring. The two affine-cover <Čech cochain complexes> for $\mathcal F$ on $C$ and for $E$ on $\mathbb P^1$ are identical. Therefore
$$
H^i(C,\mathcal F)=H^i(\mathbb P^1,E).
$$
We need an elementary presentation of any <coherent sheaf> $E$ on the <projective line>. Choose finitely many generators on each of its two standard affine charts. A generator on the first chart restricts to an element of the second chart's module localized at $t^{-1}$. Multiplication by a sufficiently large power of $t^{-1}$ brings that element into the second chart's module. Since the transition of $\mathcal O(n)$ is $t^n$, this makes the original generator extend to a <global section> of $E(n)$. A generator on the second chart similarly extends after clearing its denominator at $t=0$. Use one common exponent for all the finitely many generators. The extended sections generate $E(n)$ on both charts. Untwisting gives a surjection
$$
P=\mathcal O(-n)^{\oplus r}\longrightarrow E
$$
with coherent kernel $K$. The <long exact sequence in sheaf cohomology> and Question 3 show that $H^1(E)$ is a quotient of the finite-dimensional $H^1(P)$, since $H^2(K)=0$. Apply the same presentation argument to $K$ to obtain finiteness of $H^1(K)$. Then the segment $H^0(P)\to H^0(E)\to H^1(K)$ proves finiteness of $H^0(E)$. Higher groups vanish by Question 3. Hence \b[every $H^i(C,\mathcal F)$ is finite-dimensional.]
Next establish a useful degree estimate without using the duality requested later. A <global regular function> on $C$ is constant: a nonconstant one would give a <finite morphism> $C\to\mathbb P^1$ by Question 1. Such a morphism is surjective, because its closed image contains the <generic point>, but a regular function misses $\infty$, a contradiction. Thus $H^0(C,\mathcal O_C)=k$. Write $g=h^1(C,\mathcal O_C)$, the <genus of a smooth projective curve>. For any <divisor on an algebraic curve> $A$ and point $P$, the local <discrete valuation ring> gives
$$
0\longrightarrow\mathcal O_C(A-P)\longrightarrow\mathcal O_C(A)
\longrightarrow k_P\longrightarrow0.
$$
The last term is a length-one <skyscraper sheaf>. Additivity of the <Euler characteristic of a coherent sheaf> and repeated addition or subtraction of points give
$$
\chi(\mathcal O_C(A))=\deg A+1-g,\qquad
h^0(\mathcal O_C(A))\ge\deg A+1-g.
$$
The inequality follows because $h^1$ is nonnegative, and is sufficient for the following argument.
Fix $P\in C$. For $n\ge g+1$ this estimate gives at least two independent sections of $\mathcal O_C(nP)$, one of them the constant section. A nonconstant section is a <rational function on an algebraic variety> whose poles are confined to $P$. It must have a pole there, since a globally regular function is constant. Its extension $C\to\mathbb P^1$ is finite, and its inverse image of $\mathbb A^1$ is exactly $C\setminus\{P\}$. Consequently $U=C\setminus\{P\}$ is affine.
Let $T$ be the torsion subsheaf of $\mathcal F$ and let $V=\mathcal F/T$. The support of $T$ is finite; such a <coherent sheaf> is a finite direct sum of sheaves supported at points and has zero higher <sheaf cohomology>. The <torsion-free sheaf> $V$ is <locally free>: a finitely generated <torsion-free module> over a <discrete valuation ring> is free. Tensoring by any <invertible sheaf> is exact, so
$$
H^1(C,\mathcal F(D))\cong H^1(C,V(D)).
$$
The inclusions $V(nP)\subset V((n+1)P)$ have point-supported quotients. Their <long exact sequences in sheaf cohomology> therefore give surjections on $H^1$. Moreover
$$
\varinjlim_n V(nP)=j_*(V|_U),\qquad j:U\hookrightarrow C:
$$
a section regular away from $P$ has some finite pole order at $P$. Compute this direct limit on a finite affine cover of $C$. All its intersections with $U$ are affine by Question 2, and localization commutes with filtered direct limits. The <Čech cochain complex> consequently gives
$$
\varinjlim_n H^1(C,V(nP))=H^1(U,V|_U)=0.
$$
The maps from $H^1(C,V)$ to these groups are surjective. Choose a finite basis of this starting vector space. Each basis vector becomes zero at some stage in the direct limit; one common stage kills the entire basis. Thus there exists $n_0\ge0$ with $H^1(C,V(n_0P))=0$.
Now let $D$ be any <divisor on an algebraic curve> with $\deg D\ge n_0+g$. The estimate above gives $h^0(\mathcal O_C(D-n_0P))\ge1$. A nonzero section identifies $D-n_0P$ with an effective divisor $E$ up to <linear equivalence of divisors>. Hence $V(D)\cong V(n_0P+E)$. Adding $E$ gives a point-supported quotient of $V(n_0P+E)/V(n_0P)$, so its first <sheaf cohomology> is a quotient of $H^1(C,V(n_0P))=0$. We have proved <uniform high-degree vanishing on a smooth projective curve>:
$$
\boxed{H^i(C,\mathcal F(D))=0\quad\text{for every }i\ge1\text{ and every }D\text{ with }\deg D\ge n_0+g.}
$$
This bound concerns all high-degree divisors, not only multiples of a previously chosen divisor.
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