Is there an example of a rational polynomial that has integer output for all integer inputs that are sufficiently large? Is there an example of a rational polynomial $f(n)\in \mathbb{Q}[t]$ that has integer output for all integer inputs that are sufficiently large, but not for, say, inputs $n=1,2,3,\dots,n$?

No. If $f$ has degree $d$, you can reconstruct $f(n-1)$ from $f(n), \ldots, f(n+d)$ by taking repeated differences and adding again.