Is the Galois group associated to a random polynomial solvable with probability 0?

$G\cong S_n$ with probability $1$ as $n\rightarrow \infty$. This was proven first by

B. L. van der Waerden, Die Seltenheit der Gleichungen mit Affekt, Mathematische Annalen 109:1 (1934), pp. 13–16.

Look at this thread for more references.

Yes, the probability is zero since it equals the symmetric group with probability one:

If $Q_d(N)$ denotes the set of degree $d$ polynomials with coefficients $|a_i|\le N$ with Galois group not equal to $S_d$, then $$ |Q_d(N)|\ll N^{d-1/2}\log N $$

This bound is sufficient to prove your result. This was proven by Patrick X. Gallagher.

EDIT: Gallagher proved the following stronger result:

$$|Q_d(N)| \ll N^{d-1/2} \log^{1 - \gamma} N$$

where $\gamma \sim (2 \pi d)^{-1/2}$. Source