Correct my intuition: every Galois group is $S_n$, and other obviously incorrect statements

Question 1: Consider $f(x) = x^4 + 1$, which is irreducible over $\mathbb{Q}$. Then we may write all the roots of $f(x)$ as $\zeta_8^i$ where $i = 1, 3, 5, 7$. In particular $[F : \mathbb{Q}] = [\mathbb{Q}(\zeta_8) : \mathbb{Q}] = 4$. Thus the galois group cannot be all of $S_4$ - if you write out the automorphisms you will see that saying where $\zeta_8$ goes is the same as saying where all the $\zeta_8^i$ go.

Question 2: This was answered above, sometimes there is dependencies between the roots, so because each element of the galois group is a field automorphism, some will be disallowed.

Question 3: If $F$ is the splitting field of $f(x)$ and $f(x) = g(x)h(x)$ then if $\sigma \in Gal(F / \mathbb{Q})$ we have $\sigma(g(x)) = g(\sigma(x))$. In particular if $\alpha$ is a root of $g(x)$ then so is $\sigma(\alpha)$.

Question 4: This is a somewhat difficult question to answer. There are many criteria for when a subgroup of $S_n$ might be the symmetric group (e.g., if it is a transitive subgroup containing an $(n-1)$-cycle and a transposition). In general if I were trying to answer this in some situation I would do some algebraic number theory and look mod $p$.

Question 5: The degree $[\mathbb{Q}(a_i): \mathbb{Q}] = 4$ ($a_i$ a root of an irreducible polynomial of degree $4$, in particular $[F : \mathbb{Q}(a_i)] = 24/4 = 6$ by the tower law. You are correct, the elements are the permutations that fix $a_i$, but that is just $S_3$!.

Question 6: The element fixing $a_1$ and $a_2$ but permuting $a_3$ and $a_4$ is in $Gal( F / \mathbb{Q}(a_1))$ and is clearly a transposition.

It's important that the automorphism do more than merely move the roots around, they must be algebraically indistinguishable. Say I start with the base field of the rationals and I extend to include the solutions to $x^2+1$ which has the complex roots $\pm i$ and we can now write all numbers in this field in the form $a + bi$ in $\mathbb{Q}[i]$. I could however set $j=-i$ and then written all the numbers in the form $a + bj$ without changing anything. Complex multiplication works exactly the same way for $j$ because $j^2=-1$ just like it does for $i$.

Now compare this to field $\mathbb{Q}[\sqrt{2},\sqrt{3}]$. Lets use $j= \sqrt{2}$ and $k=\sqrt{3}$ for this example so that $j^2=2$ and $k^2=3$. If I take numbers of the form $a + bj$ and $a + bk$ then the multiplcation doesn't work the same. To see this compare $$(a+bj)(c+dj)=ac+2bd +(bc+ad)j \\ (a+bk)(c+dk)=ac+3bd +(bc+ad)k$$ and so we can see the that if I replace $j$ with $k$ the multiplication behaves differently and so this permutation is not an automorhphism. There are two automorhphisms related to sign switching as in the previous example. I can't tell $\sqrt{2}$ and $-\sqrt{2}$ apart in the above equations in the same way as in the first example and similarly for $\sqrt{3}$ and $-\sqrt{3}$. This means we have identified two non-trivial elements and they have order two. The group must have at least one other element as $2$ does not divide $3$ and these will be related to $jk = \pm \sqrt{6}$ being indistinguishable. Number in this field are all of the form $a + bj + ck + d(jk)$ and any other permutation will change the multiplication so the Galois group is the Klein four group. Permuting some of the roots gave us automorphisms but others changed the multiplication so they weren't automorphisms.

Question 1 and 2:

A $\mathbb Q$-homomorphism does not only permute the roots of some polynomial. It's also a field automorphism, and it has to obey the axioms of a field automorphism. This may restrict the choices of permutations if the roots can be written as polynomials of each other, since homomorphisms have very strong restrictions when applied to polynomial expressions. For instance, if $\alpha$ is one root of $f$, and the others are $\alpha^2,\dots,\alpha^n$, then the image of that one single root $\alpha$ under a $\mathbb Q$-homomorphism $\sigma$ already determines the images of all the other roots as well, since $\sigma(\alpha^k)=\sigma(\alpha)^k$, removing a lot of choice. An example of such a polynomial is $X^4+X^3+X^2+X+1$, whose roots are $\zeta:=e^{2\pi i/5}$ and $\zeta^2,\dots,\zeta^4$, the primitive 5th roots of unity.

Question 3:

A $\mathbb Q$-automorphism maps a root of any polynomial to another root of the same polynomial, you already got that. If a polynomial can be factorized, let's say $f=gh$, and $\alpha$ is a root of $f$, then $\alpha$ must also be a root of one of the factors (say $g$), which is where the restrictions come from: all polynomial relationships have to be preserved by $\mathbb Q$-automorphisms, not just an arbitrarily chosen one. So such an automorphism preserves both $f(\alpha)=0$ and $g(\alpha)=0$. So $\alpha$ has to be mapped to a root of both $f$ and $g$ (so essentially a root of $g$).

Question 4:

Essentially, the roots of the polynomial must not admit too many polynomial relationships between the roots (as in, one root is the image of another under a polynomial), because those would put the restrictions discussed above on the choice of permutations.

Question 5 and 6:

Why would the order be 12? If the minimal polynomial of $F/\mathbb Q$ is $(X-a_1)\dots(X-a_4)$, then the minimal polynomial of $F/\mathbb Q(a_1)$ is $(X-a_2)\dots(X-a_4)$, just drop the factor with $a_1$. The Galois group of this polynomial is at most $S_3$. In fact, taking only the subgroup of $S_4$ whose permutations fix $a_1$ is essentially the same as taking the group of permutations of only $a_2,\dots,a_4$. That's $S_3$, realized as a subgroup of $S_4$. And this realization of $S_3$ as a subgroup contains a transposition.