The original post is below. Question 1 was solved in the negative by David Speyer, and the title has now been changed to reflect Question 2, which turned out to be the more difficult one. A bounty of 100 is offered for a complete solution.

Original post. It follows from the prime number theorem and the periodicity properties $f(n+p) \equiv f(n) \mod{p}$ that for each $A < e$ there are only finitely many integer polynomials $f \in \mathbb{Z}[x]$ such that $|f(n)| < A^n$ for all $n \in \mathbb{N}$. On the other hand, for each $k \in \mathbb{N}$ the binomial coefficient $\binom{n}{k}$ is an integer-valued polynomial in $n$ bounded by $2^n$.

Question 1. Are there infinitely many integer polynomials with $|f(n)| < e^n$ for all $n \in \mathbb{N}$?

Question 2. Given $A < 2$, are there only finitely many integer-valued polynomials $f \in \mathbb{Q}[x]$ with $|f(n)| < A^n$ for all $n \in \mathbb{N}$?

Question 2:
The constant $A$ can be brought down to $\sqrt 3$, and probably
a bit below that but not all the way down to $1+\epsilon$.

Instead of the polynomial $f(n) = {n \choose m}$, use a
finite difference of such polynomials, 
$$
f(n) = \sum_{i=0}^m (-1)^i {m \choose i} {n \choose m+i}.
$$
This is the $x^n$ coefficient of 
$$
\frac1{1-x} \left( \frac{x(1-2x)}{(1-x)^2} \right)^m
$$
and can be estimated by contour integration on $|x| = 3^{-1/2}$;
the maximum of $|f(n)|^{1/n}$ occurs near $n=3m$, for which
the critical points are at $x = (3 \pm \sqrt{-3}) / 6$.

Note that for $f(n) = {n \choose m}$ the generating function was
$\frac1{1-x} (x/(1-x))^m$, and the maximum occurred near $n=2m$, 
for which the critical point was at $x = 1/2$.
The factor $1-2x$ kills that maximum, and it seems that
using $x/(1-x)$ and $(1-2x)/(1-x)$ to the same power is optimal.
To reduce $A$ further, try to include also some power of $(3x^2-3x+1)/(1-x)^2$
to kill the new critical point.
