Maybe it should be added that this book of Lang's is not at all held in high esteem by the specialists, who seem to find it out of focus on the main issues. I don't have an opinion of my own on this.

One moral: study of an abstract finite group and its representations (characters, Schur indices) requires a distinction between the given group and its more concrete homomorphic or isomorphic images in groups like $GL_n(\mathbb{C})$. These matrix groups may or may not be generated by pseudoreflections, might be isomorphic without being conjugate, etc. Such matters always come up for instance in crystallography. In the original question, the given abstract groups are dihedral but could be other finite Coxeter groups (= real reflection groups), etc.

Two clarifications of the language you use here: 1) "complex representation which is generated by pseudoreflections" should be "complex representation of a group generated by ..."; 2) dihedral groups are actually real reflection groups, to which the more general theory of pseudoreflection (or complex) reflection groups is applied in the context of invariant theory, since this bigger class of groups is characterized by having a polynomial ring of invariants in the natural representation. Jack's answer based on induction is best here, but anyway your last question should be restated.

The general theory goes back a century and is treated in many books. Study of particular groups and representations is more of an art form, e.g., for simple groups. One helpful textbook source is Serre (Springer GTM 42): dihedral group representations are written down as induced representations in 5.3; later the Schur index is discussed in 12.2 (note Exercise 12.1). Books by Curtis & Reiner also have full treatments.

While the abstract Cartan is the correct gadget to introduce, the original question can be dealt with concretely if it's made clear up front what you mean by "induction". The chosen Borel subgroup itself and not just its characters must play a role in the process of induction. Any of its maximal tori, or the abstract one, will do here, but $B$ determines the positive roots and hence the dominant weights. (No assumption yet on characteristic, until you ask for irreducible representations.)

At the opposite extreme, a finite simple group of Lie type can't exhibit this odd numerical behavior due to the existence of Steinberg characters (and the fact that class sizes divide the group order in general). I have no idea about other non-nilpotent cases, but I suspect Marty Isaacs (Wisconsin) would know how to answer the original question even without the help of Magma if he was motivatedto do so. In any case, Kevin is correct to start with nonabelian 2-groups since these are the prime suspects for counterexamples and there are a lot of them.

Since this section of Artin's book is concerned with approximating the gamma function, there may be some tendency to use the equals sign loosely. I guess the point is to find a convenient elementary function giving a good approximation for large $x$; the choice might be fine-tuned in various ways. But in the era before computers the shape of an approximating function would have been the most interesting question for many people.

A little more detail about [AJS], which is a 300+ page paper (Asterisque 220, 1994) but has a readable introduction. They work in a graded setting with Lie algebra modules, getting for large enough $p$ and suitably bounded weights a precise comparison with the quantum enveloping algebra of Lusztig at a $p$th root of unity. In the latter case, one combines work of Kashiwara-Tanisaki on the analogue of the Kazhdan-Lusztig Conjecture for affine Lie algebras with work of K-L passing from there to quantum groups. [AJS] relies more on combinatorics than on geometry.

A later paper claims to find groups not found by Blichfeldt, but I haven't looked into this myself: MR1169227 (94b:14045) Yau, Stephen S.-T.(1-ILCC); Yu, Yung(RC-TAIN) Gorenstein quotient singularities in dimension three. Mem. Amer. Math. Soc. 105 (1993), no. 505, viii+88 pp. This kind of list-making is often error-prone, so it would be helpful to have a list everyone agrees on.

In general Kostant's approach to weight multiplicities showed that these typically grow very large, bounded only by the Kostant partition function value for the difference between highest weight and lower dominant weight. But the inverse matrix is elusive. Richard's suggestion that entries might be relatively small in absolute value for type $A$ could carry over (at least qualitatively) to other Lie types, though it's hard to visualize the combinatorics for say type $E_8$.

Hyman Bass and his students (including Anthony Bak, Michael Stein) in the early 1970s did a lot of work on $K_2$ involving classical groups besides those of type $A$, before Quillen's success with the higher $K$-groups. The typewritten lecture notes by Bass Algebraic K-Theory (W.A. Benjamin, 1968) contain good foundational material but are now hard to find and naturally aren't up to date.

Your questions cover a lot of territory, so there is by now a lot of literature including extensive work on the congruence subgroup problem and on algebraic K-theory. It might help to separate the questions and give a little more context.

You definitely have to be more careful when working with possibly nonreduced group schemes. Standard references include the book by Demazure and Gabriel; see Part I, Chapter 5 in Jantzen's Representations of Algebraic Groups for a development in this spirit taking into account Frobenius kernels, etc. For reduced groups, the characteristic of the field doesn't really matter: see Chapter II in Borel's Linear Algebraic Groups (or similar material in books with the same title by Springer and me).

Probably the best you can do online is to type in a search word like "supersoluble", which should return many but not all pages of Platonov's article: e-math.ams.org/bookstore-getitem/item=TRANS2-69 I've never explored this area of the literature on algebraic groups, which got developed by algebraists in the former Soviet Union influenced by abstract group theory. So I'm not sure what's really there.