What can we say about the differences between roots of a polynomial with large Galois group?

Suppose that $K$ is a number field and $L$ is the splitting field of a monic polynomial in $\mathcal{O}_{K}[x]$ of degree $d \geq 5$ with roots $\alpha_{1}, ... , \alpha_{d}$. Assume that the $\mathrm{Gal}(L / K)$ is the full symmetric group $S_{d}$ (there is an obvious variant of this question where we assume that the Galois group is $A_{d}$ instead).

Vaguely stated, my question is, given the relative lack of algebraic relations among the roots, what can one conclude about whether there exist primes $\mathfrak{P}$ of $L$ such that certain subsets of the roots become equal modulo $\mathfrak{P}$? More specifically, could one conclude, for instance, that there exists a prime $\mathfrak{P}$ of $L$ such that $\alpha_{i} - \alpha_{j} \in \mathfrak{P}$ for exactly one choice of $\{i,j\} \subset \{1, ... , d\}$?

Here is a (possibly) closely related question: what can we say about the powers of primes of $K$ which contain the discriminant $D$ of this polynomial (again assuming its Galois group is $S_{d}$)? Can one conclude that there exists a prime $\mathfrak{p}$ of $K$ such that $D \in \mathfrak{p} \setminus \mathfrak{p}^{2}$? Is it possible for $D \in (K^{\times})^{n}$ for some $n \geq 3$?

I'm sorry that I'm asking several vaguely related questions here rather than narrowing things down to be more concrete. But I've been trying but failing to get small results on this theme using elementary algebraic number theory for a while now, and I'm curious as to whether any statements of this kind are already known. (Perhaps unsolvability and $2$-transitivity are the only properties of $S_{n}$ which we really need to use here.)

Kedlaya proved that for every $n$ there is a monic polynomial $f\in\mathbb Z[X]$ with square-free discriminant and Galois group $S_n$. See here (published version) or here (preprint).