In higher dimension, since $L_0$ is ample, you can take a surface $Y \subset X$ through $C$ in the linear equivalence class $L_0^{g-2} \in \mathrm{CH}^2(X)$ (i.e., linearly equivalent to a multiple of a $2$-plane in the appropriate projective embedding). Since $L_0^{g-2}.L^2 > 0$ we still have $L_{|Y}^2 > 0$, and the same argument shows that $L.C < 0$ contradicts the Hodge index theorem on the surface $Y$.

@ Noam Elkies: Thank you! It was of indeed the example of $\binom{n}{k}$ which led me to ask question 2. (Otherwise I was considering rather different kinds of stuff, more in line with the first question.)

There is in fact the more precise finiteness theorem of Chow, which states that the set of all closed subvarieties of $\mathbb{P}^n$ of a bounded degree is contained in only finitely many deformation types. Its proof for curves is particularly easy.

(cont.) This generalizes in an obvious way to global fields and to algebraic curves of higher genus. It would thus be a sharp algebraicity criterion, of which I can only prove a weaker version involving an inequality stronger than 1). It was this type of problems which motivated me to ask question 1.

(cont.) is a rational fraction of degree $\leq h_p$. Then: 1) If $\log{\rho} + \liminf_n \frac{1}{n} \sum_{p : h_p < n} \log{p} > 0$, the set $S(G,h)$ should only contain rational functions in $\mathbb{Z}[[x]]$. 2) If however the inequality in 1) is not satisfied, the set $S(G,h)$ should be uncountable.

Here is a generalization of Ruzsa's conjecture which, though not involving polynomial coefficients anymore, is nonetheless in the spirit of the two questions. It is suggested to me by the connection with Fekete's theorem (and by that of Polya-Carlson-Bertrandias). Let $G \ni 0$ be a pointed (say) simply connected domain, and $\rho$ its conformal mapping radius. Let $h : P \to \mathbb{N}_0 \cup \{\infty\}$ be a function on the set $P$ of primes, and consider the set $S(G,h)$ of those $f \in \mathbb{Z}[[x]]$ which are meromorphic on $G$ and whose reduction $f \mod{p}$ at each prime $p$ (cont.)

@ Noam Elkies: I think so; the relation to Fekete's theorem which you and David discovered seems particularly interesting to me. My contribution here, though, is minimal and reduces to having merely asked the question.

Well, actually, he would have to donate it to you - for I already awarded you the old 100 bounty. Then I intended to set a new 100 bounty for him, but it turned out this wasn't an option, so I went for 200. (The latter hasn't yet been awarded, so if I award it to you instead, what you say could be an option if you set a 150-point bounty, if that's possible, for his answer :).

A bounty of 200 will be awarded to this answer in 24 hours. (It turns out that 1) one may not set two bounties of the same worth on the same question, and 2) one has to wait for 24 hours before the bounty can be awarded.)

In any case, this is amazing. Now I have to award the bounty. Since the solution is a joint one, here is what I will do. I will award the current bounty to Noam Elkies' answer, and start a new bounty of the same worth to reward this complete answer.

A generalization of Fekete's theorem inspired by arithmetic geometry (and, I think, in a particularly illuminating way) appears in Chinburg's paper "Capacity theory on varieties": archive.numdam.org/ARCHIVE/CM/CM_1991__80_1/CM_1991__80_1_75‌​_0/… See Theorem 1.2 and the Minkowski application in the paragraph following it.

(And I meant, after all, to write $\mathbb{Z}[1/(1-x)] \subset \mathbb{Z}[[x]]$ and $f \in \mathbb{Z}[[x]]$ in the above comment. Sorry about this, as well as for the switch of notation, $f$ now denoting the generating function.)

(continued.) And a closer look at their argument shows that not only the order of the differential equation can be taken to depend only on the radius of convergence, but that if the coefficients satisfy $|a(n)| < A^n$ for all $n \geq n_0$ and $A < e$, then in fact $f$ satisfies one of a finite set (depending on $A$ and $n_0$) of linear homogeneous differential equations. This implies that the degree of a polynomial in Question 1 is bounded for $A < e$.

Not an application, but it came up in relation to an old conjecture of Ruzsa characterizing the subring $\mathbb{Z}[1/(1-x)] \subset \mathbb{Q}[[x]]$ by the two properties: 1) radius of convergence (and I would add: meromorphy) is strictly $> 1/e$; and 2) for a set of primes $p$ of full density, the mod $p$ reduction is $A_p(x)/(1-x)^p$, with $A_p$ a polynomial of degree $< p$. Perelli and Zannier have shown that such an $f \in \mathbb{Q}[[x]]$ is at least $D$-finite. (continued.)

Thanks also to David Speyer for his solution to Question 1. But the bounty was offered for Question 2, so I'm marking this as the accepted answer.