I've read of Chebotarev Density Theorem. The statement over $\mathbb{Q}$ is:
Let $K$ be Galois over $\mathbb{Q}$ with Galois group $G$. Let $C$ be a conjugacy class of $G$. Let $S$ be the set of (rational) primes $p$ such that the set of $\{\text{Frob}_{𝔭} | 𝔭 \ \text{above } p\}$ is the conjugacy class $C$. The set $S$ has density $\frac{|C|}{|G|}$.
I understand the statement, and my question is not about that, but the relation of this statement with facts about polynomials. There is the following fact, which tells that there is a correspondence:
Let $K = \mathbb{Q}(\alpha)$, where $\alpha$ is an algebraic integer. Let $f$ be the minimal polynomial of $\alpha$, and let $L$ be the Galois closure of $K$ with Galois group $G$. Let $p$ be a (rational) prime which does not ramify. The factorization of $f$ modulo $p$, that is, $f \equiv f_1f_2 \ldots f_k \pmod{p}$, is the same as the cycle type $(f_1, f_2, \ldots, f_k)$ of the $Frob_𝔭 \in G$ permuting $f$'s roots, where $𝔭 \in L$ is a prime ideal above $p$.
(I would also be happy to see a reference/proof to this theorem, but that's not the main issue. I'm also interested to know if there is some other way to motivate the connection between Frobenius maps and polynomials)
Now, for the questions.
Can we apply the Chebotarev Density Theorem for determining the density of a given factorization type for a given polynomial? For example, if we have the polynomial $x^4 + x + 1$, can we calculate how often we have the factorization type $(1, 3)$. This seems possible to me using the other fact I wrote, but I am not able to fill in the details.
Moreover, can this factorization type be reached from more than one conjugacy class? That is, can there be two (or more) conjugacy classes $C_1, C_2$ such that the $p$ with $\sigma_p = \{\text{Frob}_𝔭 | 𝔭 \text{ above } p\} \in C_1, C_2$ have the cycle type $(1, 3)$?
Does the situation in 1. change for reducible polynomials - is the calculation still possible?
I also have a loosely-related question about applying the density theorem to obtain some other results:
4. In http://websites.math.leidenuniv.nl/algebra/Lenstra-Chebotarev.pdf there is an example application of the Chebotarev density theorem for proving the Dirichlet's theorem on arithmetical progressions at page 4 in the paragraph starting with "If you apply this theorem in the abelian case, ..."
In general, I'm very confused about the proof. For me the most intuitive way to approach the problem would be looking at the cycle types of $\sigma_p$, but the proof seems to "cheat" by looking at $\sigma_p$ as integers modulo $m$. There seems not to be a polynomial whose factorization we would inspect, which confuses me - how does this other method precisely work, what happens in the step $\sigma_p \longleftrightarrow (p \text{ mod } m)$? I kind of feel like I've answered my own question already, but if someone has some clarifying ideas, please share.