Yes, this is the standard notion developed originally in the setting of finite dimensional irreducible representations of complex semisimple Lie algebras (or groups); extremal weights have multiplicity 1. This can to some extent be carried over to "integrable" representations of affine Kac-Moody algebras, where Kac found a good analogue of the Weyl character formula. Also, the theory of Demazure modules (geometrically motivated) involves study of the subspace obtained by fixing an extremal weight space and applying to it all negative root vectors in the Lie algebra.

It's worth noting that the questions here make equal sense and have mostly the same answers when the group is assumed to be a (connected) semisimple algebraic group over an arbitrary algebraically closed field. In any case, smaller fields need more discussion.

@Pasha: Probably the arXiv labels will need tweaking over time. Maybe add an LT (Lie theory) option? But like MathSciNet it is tricky to devise permanent classifications, since those that fall into disuse can't be recycled like old telephone numbers without messing up searches. My problem with arXiv is that I don't have time or energy to browse daily the bigger lists (CO, GR, RA, QA, etc.) so I focus on a few labels and trust people to cross-reference carefully when they post. This blog invents lots of new tags, maybe too many to keep track of.

As noted, the question is about Hopf algebras and needs such a tag. The concept originates with work of Earl Taft (retired from Rutgers) and began to show up in arXiv preprints such as math.QA/0009214. But to get back to the origins one should search earlier using MathSciNet if possible. Searches on Google, Wikipedia, or even arXiv are probably too limited for this purpose.

The question needs better focus, I think. (More context and motivation would also help.) There is no obvious connection here with reflections or with Dieudonne's theorem: the square of your $s_t$ is not a "reflection" and no bounds on number of generators are implicit here. The type of field (finite, etc.) plays a role, along with the $k$-structure. Generation of semisimple groups comes up in work of Steinberg, Tits, and others; for more general groups you want to distinguish cases where a Levi decomposition exists and those in characteristic $p$ where it doesn't, etc.

This is a helpful reference, since de Graaf is well grounded in the subject matter and has a lot of computational expertise as well. Note that his arXiv preprint is posted only in v1, while the published paper has an expanded reference list and presumably a number of other changes (he thanks the "referees" for improving the exposition). Note too that the arXiv subject is math.RA, which unfortunately I don't routinely consult. This is one case where the 17B in MathSciNet works better for me; but no system is perfect.

P.S. The tag algebraic-groups really belongs here, along with others.

For the Borel fixed point theorem, see a 1977 note by Steinberg giving a streamlined approach (reprinted by AMS in his collected papers): MR0466336 (57 #6216), Steinberg, Robert, On theorems of Lie-Kolchin, Borel, and Lang. Contributions to algebra (collection of papers dedicated to Ellis Kolchin), pp. 349–354. Academic Press, New York, 1977. For homogeneous spaces, I don't know a real shortcut in arbitrary characteristic where the results belong and tangent spaces (but no Lie algebras) occur. P.S. Dan Allcock found shortcuts to rank one theory: arXiv 2007, then J. Algebra (new title).

The "five" methods listed aren't all distinct (and the characteristic of the field shouldn't matter). It's essential here to put the question in the context of what one already knows and what is the overall goal. Most treatments originate in the 1956-58 Chevalley seminar dealing with structure and classification of semisimple groups, followed by some evolution in exposition and a few simplifications. Shortcuts for general or special linear groups may be useful for some purposes, but the general result takes some work even though it's "elementary".

The added reference is helpful. It's a somewhat amended version of the paper published in Comm. Math. Helv. 72 (1997), 503-520 and appears on Soergel's homepage: home.mathematik.uni-freiburg.de/soergel

This is a useful reminder that countless dissertations around the world remain formally unpublished. Many of these do lead to papers, but for example Ming-Peng Gong has only some papers up to 1997 listed by MathSciNet. Many sources also contain small or large errors, even after being refereed. List-making in mathematics is popular but especially error-prone: e.g., Killing was the first to find all the simple Lie algebras over $\mathbb{C}$, but thought there were two versions of $F_4$. Is there a 100% reliable published list of nilpotent Lie algebras of dim $\leq 6$? I don't know.

Probably you want to require your modules to be finitely generated. In any case, it's a good idea to look at a short paper by Soergel in which he notes both positive and negative results closely related to the question you raise: MR0872544 (88c:17011) Soergel, Wolfgang, Équivalences de certaines catégories de ${\germ g}$-modules. (French. English summary) [Equivalences of certain categories of $\mathfrak{g}$-modules], C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 15, 725--728.

One person's "tedious" is another person's "fascinating". Much of the appeal in the classification of finite simple groups lies in the individuality of the groups which mysteriously appear in this list.