Harish-Chandra modules have by now been studied extensively in a fairly algebraic context by Joseph (in many papers), Jantzen (especially in his enveloping algebra book, written in German), and many others. Even if your motivation comes from unitary representations of real Lie groups, as in work of Vogan and others, you are led to complexify the Lie algebras. The BGG category O theory illuminates mainly the complex Lie group part of the theory, but my book just tells the algebraic story leading up to the use of geometric/homological methods. This is already a long story.

Yes, this is the way it was understood originally. Even though the papers assume characteristic 0, the ideas carry over well to any algebraically closed field. But in later literature more emphasis was placed on describing centralizers of semisimple elements as in Chapter 2 of my 1995 book and the sources cited. This constructs subgroups of maximal rank less directly than you want. It's hard to locate a reference for your Lemma 2 short of SGA 3. (But see Bourbaki, VI.1.7 on closed symmetric sets of roots, especially Prop. 23.)

This theorem from SGA 3 is an efficient reference for Lemma 2, though the background and setting are more than needed to describe the structure theory over an algebraically closed field. Concretely, the desired subgroup is generated by a maximal torus together with pairs of root subgroups (or the rank 1 semisimple groups they generate). The standard commutation relations show that no further roots appear if the given set of roots is closed and symmetric. Here as in SGA 3, the Lie algebra reflects accurately what is going on since the subgroup is of maximal rank.

Yes, I did lose the title after reading the question. This can be formulated in a number of settings: complex semisimple Lie groups or Lie algebras or compact groups or algebraic groups in any characteristic. Even formulated narrowly, the question is not easy to answer definitively. If it were, invariant theory might be less challenging.

I will look further at literature later this weekend, but meanwhile I think the essential point coming from the 1949 Borel-de Siebenthal paper (as generalized to the algebraic group situation in any characteristic) is that the connected reductive subgroups of maximal rank are generated by a maximal torus along with root subgroups belonging to a root system found via the extended Dynkin diagram. Sets of simple roots only generate the root systems in Levi subgroups; the others are usually called "pseudo-Levi", like the copy of $A_2$ inside $G_2$.

It seems to be understood here that everything is done in characteristic 0 (in characteristic p it all gets much worse). To underscore Ben's point, you could start with the special linear group $G$ and embed into it an arbitrary connected semisimple group $H$. Then almost nothing can be said about restricting representations. Even if $H$ is generated by pairs of one dimensional root subgroups relative to a given maximal torus of $G$, branching rules are usually quite intricate to work out. For invariants, maybe the papers of Roger Howe et al. would help?

You should try arXiv, for example arXiv:0905.1282, and perhaps ask Ruth Charney (Brandeis) or look at her recent papers on artin groups.

For a bigger picture still, look at: MR2373153 (2009a:55012) 55R35 (20F55 22E40 55P35 57S15) Andersen, K. K. S. (DK-ARHS); Grodal, J. (1-CHI); Møller, J. M. (DK-CPNH); Viruel, A. (E-MAL) The classification of p-compact groups for p odd. Ann. of Math. (2) 167 (2008), no. 1, 95–210

There is interesting later work on this kind of question for compact Lie groups and analogues, for example by Dwyer and Wilkerson. A follow-up to the paper by Tits ("part I" still searching for its "part II"), with useful references: MR2174268 (2006f:55015) Dwyer, W. G.; Wilkerson, C. W. Normalizers of tori. Geom. Topol. 9 (2005), 1337--1380

Footnote: "MathSciNet search does not reveal anything in this topic" is not surprising, since the search function of MathSciNet has built-in limitations. The actual content of papers cannot be searched, so results depend on what happens to be written in reviews along with titles of papers and subject numbers. No one has yet invented an all-knowing search engine to scan all of the expanding mathematics literature for a particular thought. So humans may still have a minor role to play in answering questions. (For a while anyway.)

Notational detail: usually people write either $GL(n,\mathbb{C}) or more functorially, $GL_n(\mathbb{C})$. The original post uses a highly nonstandard form.

In general, it's hard to say anything helpful about the decomposition of an induced module or of a tensor product of two modules for a finite group. For groups of Lie type, much work has been done in such directions for specific types of subgroups and modules, but results usually depend strongly on special methods such as Hecke algebras. Clifford theory and Mackey theory do provide some general guidelines for arbitrary finite groups, of course.

Be careful about the dimensions involved: induction multiplies the dimension of a given module by the index $[G:H]$, so your second module is typically bigger than your first.