Let $f(x) \in \mathbb{Z}[x]$ be a polynomial. Let $f_p(x) \in \mathbb{Z}_p[x]$ denote the polynomial $f \bmod{p}$ (where $p$ is a prime). We say $p$ is good for $f$ if $f_p(x)$ splits (into linear factors) over $\mathbb{Z}_p$. We also say that $p$ is $d$-good if there exists a constant $d$ such that $f_p(x)$ splits into irreducible factors of degree at most $d$ over $\mathbb{Z}_p$.
$f(x) \in \mathbb{Z}[x]$ is said to be low degree if its degree is smaller than $n$ for some parameter $n$. A prime is said to be small if it is $O(\log{n})$ bits long.
Now:
(a) Is there a family of small primes which is good for all low degree polynomials?
(b) Barring which, what fraction of small primes are good for a fixed low degree polynomial?
(c) Barring which, what fraction of small primes are $d$-good (preferably $d = 2$) for a fixed low degree polynomial?