= Solution
Write the <Taylor series> of the entire function as
$$
f(z)=\sum_{k=0}^{\infty}c_kz^k.
$$
The <Cauchy estimate> on the circle $|z|=R$ gives
$$
|c_k|\leq\frac{\max_{|z|=R}|f(z)|}{R^k}
\leq aR^{n/2-k}+bR^{-k}.
$$
If $k>n/2$, letting $R\to\infty$ gives $c_k=0$. Since $n$ is odd,
$$
\boxed{\deg f\leq\lfloor n/2\rfloor}.
$$
For the second question, suppose such an $f$ existed. It never vanishes, so $g=1/f$ is analytic on $\mathbb C\setminus\{0\}$ and
$$
|g(z)|\leq\sqrt{|z|}.
$$
The <Riemann removable singularity theorem> extends $g$ analytically across zero with $g(0)=0$. Applying the result just proved with $n=1$, $a=1$, and $b=0$ makes $g$ a polynomial of degree at most zero. It must then be identically zero, contradicting $g=1/f$ away from zero. Therefore no such function exists.
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