The answer to Question 1 is yes, although it's a bit annoying to extract an explicit bound from the argument I have in mind. First observe that the question reduces to a question about largest (in absolute value) roots of monic integer polynomial of degree $n$ with constant term $1$: namely, it's equivalent to asking whether the smallest possible such roots greater than $1$ in absolute value are bounded away from $1$ in absolute value, with the bound possibly depending on $n$.

If such a polynomial has $k^{th}$ coefficient at least ${n \choose k} r^k$ in absolute value then it must have a root of absolute value at least $r$. It follows that for fixed $r$ and $n$, the set of such polynomials with all roots of absolute value at most $r$ is finite, so the conclusion follows.

The answer to Question 1 is yes, although I don't think I can extract a reasonable bound from the argument I have in mind. First observe that the question reduces to a question about largest (in absolute value) roots of monic integer polynomial of degree $n$ with constant term $1$: namely, it's equivalent to asking whether the smallest possible such roots greater than $1$ in absolute value are bounded away from $1$ in absolute value, with the bound possibly depending on $n$.

If such a polynomial has $k^{th}$ coefficient at least ${n \choose k} r^k$ in absolute value then it must have a root of absolute value at least $r$. It follows that for fixed $r$ and $n$, the set of such polynomials with all roots of absolute value at most $r$ is finite, so the conclusion follows.