If you take the splitting field of $x^5+ax+b$ and consider it as an extension of its quadratic subfield, then it will be unramified with Galois group contained in $A_5$ whenever $4a$ and $5b$ are relatively prime. This is a result of Yamamoto. For almost all $a$ and $b$ (specifically, on the complement of a thin set), the group is $A_5$.
You might also enjoy this preprint of Kedlaya, which I found very readable. A note on Kedlaya's webpage, dated May 2003, says that he will not be publishing this because it has been superseded by a recent result of Ellenberg and Venkatesh. I assume he is referring to this paper, but I can't figure out why that one supersedes his.
Perhaps you should separate this into at least two questions, as your first two questions are "local" and your last three are "global".
As to the local ones:
This response is more an idea than a complete answer. But it seems to me that the Krasner-Serre mass formula should tell you whatever you want to know about the number of extensions of a $p$-adic field of given degree. (Note that it comes down to counting totally ramified extensions, since any other kind is much easier to count.) Casting about just now for a good reference, I looked through my own course notes on local fields and was severely disappointed: I say too little and what I do say is riddled with typos. But this paper of Pauli and Roblot seems to be, among other things, a very thorough survey of these $p$-adic mass formulas. In particular it contains references to the original papers of Krasner (1966) and Serre (1978).
I haven't looked at the details myself, but surely (meaning, of course, that I am not completely sure!) this mass formula will answer your first question. It also seems to have a good chance to answer your second question, possibly along with some inclusion-exclusion/Mobius inversion arguments.
As to the global ones:
3) Yes! This is a famous theorem of Hermite. Look in a good algebraic number theory book, e.g. one written by Neukirch.
4) I don't, no, off the top of my head, but others surely do. Stay tuned...
5) I guess I don't see why this should be true, but I'll have to think more about it.
As above, asking fewer questions at a time will probably elicit more detailed answers.
Best Answer
$\DeclareMathOperator\Gal{Gal}$The answer is "yes", and this is an easy exercise in class field theory: if, for example, $q$ is a prime number that is $1\pmod n$, and $F$ is a quadratic number field in which $q$ splits, then there is a quotient of the ray class group of $F$ with modulus $q$ that has order $n$ and on which $\Gal(F/\mathbb{Q})$ acts by $-1$, so that the corresponding ray class field $L/F$ is Galois over $\mathbb{Q}$ with dihedral Galois group and is unramified (over $F$) outside of $q$. See, for example, Section 3.1 in https://arxiv.org/abs/0805.1231 for the details of this computation. So if you also arrange $F$ to be unramified at all $p|2n$, then $L/\mathbb{Q}$ is as required.