The extension $H/\mathbf Q$ is never abelian, unless $H=\mathbf Q(\zeta_q)$ (which happens for just a few values of $q$, since the class number of $\mathbf Q(\zeta_n)$ grows very fast). Indeed, according to Kronecker-Weber, every abelian extension of $\mathbf Q$ is contained in a cyclotomic extension. If $H/\mathbf Q$ were abelian, we would have $\mathbf Q(\zeta_q) \subseteq H \subseteq \mathbf Q(\zeta_n)$ for some $n$; then, examining the ramification, we see that $n=q$ and so $H=\mathbf Q(\zeta_q)$.
The degree of $H/\mathbf Q$ is equal to $q-1$ times the class number $h_q$.
In general, the primes which split completely in the Hilbert class field are those which are principal in the base field.
The class group of $\mathbf Q(\zeta_n)$ is a very complicated thing. Whole books are devoted to it. It is a fascinating topic with deep connections to the theory of the Riemann zeta function.
I recommend Washington's book Introduction to Cyclotomic Fields. If you are interested specifically in Iwasawa theory, I would recommend the book Cyclotomic Fields and Zeta Values by Coates and Sujatha, which is a great introduction to this truly unbelievable theory.
Here are some answers to your questions. Throughout, fix a number field K and a finite S of primes of K (one could also take infinite sets of primes by inductively enlarging finite S ’s, but this is a detail).
1) For short, a finite extension L/K is called S-ramified if it is unramified outside S. The composite of two such extensions is again S-ramified, so it makes sense to consider the composite $K_S$ of all the finite S-ramified extensions of K. Because of maximality, $K_S/K$ is obviously Galois ; its Galois group $G_S (K)$ is profinite. The quotient $G_S (K)^{ab}$ of $G_S (K)$ modulo the closure of its commutator subgroup fixes the maximal S-ramified abelian extension $K_S^{ab}$ of K. This $K_S^{ab}$ can also obviously be defined as the composite of all the finite abelian S-ramified extensions of K. For more details on ramification in infinite Galois theory (although there is no mystery), see e.g. §2 of the appendix of Washington’s « Introduction to cyclotomic fields ».
2) Actually, class field theory (CFT) also takes into account the ramification at infinite primes (i.e. embeddings into an algebraic closure) in a finite extension L/K : by definition, a complex prime is always unramified ; a real prime of K is ramified iff it becomes complex in L. A « K-modulus » M is a formal product of an integral ideal of K and a finite number of distinct real embeddings. The classical formulation of CFT in terms of ideals and modulii is not very suggestive at first sight. Let us refer for this to the excellent survey by D. Garbannati, « CFT summarized », Rocky Mountain J. of Math., 11, 2 (1981), 195-225. Then the ray class field modulo M over $K$ is the largest finite abelian M-ramified extension of K. If $S = S_p$ := the set of primes of $K$ above an odd prime $p$ (to avoid petty details on ramification at infinite primes), indeed $K_S^{ab}$ is the direct limit of the ray class fields mod $p^n$ as you suspected.
3) The nowadays prevailing formulation of CFT in terms of idèles is well-adapted to the machinery of Galois cohomology and Poitou-Tate duality. But even so, the description of $G_S (K)^{ab}$ is explicit in particular cases (if $K = \mathbf Q$ , see the Kronecker-Weber theorem), but not in general. Actually many deep conjectures on the determination of $G_S (K)^{ab}$ and far reaching applications remain unsolved to this day. Let us for instance take the p-adic point of view and study the maximal pro-p-quotient $X_S (K)$ of $G_S (K)^{ab}$ . If S contains $S_p$, CFT tells us that $X_S (K)$ is a $\mathbf Z_p$-module of finite type, hence it is a priori of the form $T_S (K)$ x $\mathbf Z_p^{1+c+d}$, where $T_S (K)$ is a finite abelian p-group, $2c$ is the number of complex primes of $K$ and $d$ is a natural integer. The celebrated Leopoldt conjecture (up to now proved only for abelian $K$) predicts the nullity of $d$. If $K$ is totally real, then $T_S (K)$ contains, in a precise sense, the theory of $p$-adic $L$-functions : this is the « Iwasawa main conjecture », proved by Mazur-Wiles and Wiles. If $S$ does not contain $S_p$, much less is known ./.
Best Answer
$\Bbb Q(\sqrt{-5})$ has class number two, so its Hilbert class field is a quadratic extension. That quadratic extension is $\Bbb Q(i,\sqrt5)$.