Borel's lengthy 1953 Annals paper is essentially his 1952 Paris thesis. It was
followed by work of Bott, Samelson, Kostant, and others, which eventually answers your
side question affirmatively. For a readable modern account in the setting
of complex algebraic groups rather than compact groups, try to locate a copy of the lecture notes: MR649068 (83h:14045) 14M15 (14D25 20F38 57N99 57T15)
Hiller, Howard,
Geometry of Coxeter groups.
Research Notes in Mathematics, 54.
Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. iv+213 pp.
ISBN 0-273-08517-4. (This was based on his 1980 course at Yale. Eventually
he left mathematics to work for Citibank.) The identification of the cohomology ring with the coinvariant algebra of the Weyl group has continued to be important for algebraic and geometric
questions, for instance in the work of Beilinson-Ginzburg-Soergel. While Hiller's notes are not entirely self-contained, they are helpfully written. (But note that his short treatment of Coxeter groups has a major logical gap.)
ADDED: In Hiller's notes, Chapter III (Geometry of Grassmannians) is most
relevant. For connections with Lie algebra cohomology, the classical paper
is: MR0142697 (26 #266) 22.60 (17.30)
Kostant, Bertram,
Lie algebra cohomology and generalized Schubert cells.
Ann. of Math. (2) 77 1963 72–144. Nothing in this rich circle of ideas can be made
quick and easy; a lot depends on what you already know.
P.S. Keep in mind that Hiller tends to give explicit details just for the
general linear group and grassmannians, but he also points out how the main
results work in general, with references. I don't know a more modern textbook
reference for this relatively old work. But the intuitive connection between
the Borel picture and the Bott/Kostant cohomology picture is roughly this: The
Lie subalgebra spanned by negative root vectors plays the role of tangent space
to the flag manifold/variety. In the Lie algebra cohomology approach you get an explicit graded picture for each degree in terms of number of elements in
the Weyl group of a fixed length, whereas the Borel description in terms of Weyl group coinvariants makes the
algebra structure of cohomology more transparent. (What I
don't know is whether a simpler proof of Borel's theorem can be derived using Lie algebra cohomology.)
Concerning the relationship between $K/T$ and $G/B$, this goes back to the
work around 1950 on topology of Lie groups (Iwasawa, Bott, Samelson): all the
topology of a connected, simply connected Lie group comes from a maximal compact subgroup. So the two versions of the flag manifold are homeomorphic.
In later times, emphasis has often shifted to treating $G$ as a complex algebraic group, so that $G/B$ is a projective variety. For me the literature is hard to compactify.
One more reference, which treats the Borel theorem in a semi-expository style: MR1365844 (96j:57051) 57T10
Reeder, Mark (1-OK),
On the cohomology of compact Lie groups.
Enseign. Math. (2) 41 (1995), no. 3-4, 181–200. There is some online access here.
If i have correctly understood your question, there are various such examples, arising from mathematical physics contexts (where some of the original motivations for the study of quantum groups first appeared).
One such example, is the case of the $q$-Heisenberg algebra: If we consider the (usual) 3d Heisenberg Lie algebra $L_H$, generated by $a,a^{\dagger},H$ subject to the relations:
$$
[a,a^\dagger]=H, \ \ \ \ \ \ \ \ \ [H,a]=[H,a^\dagger]=0
$$
then the $q$-deformed Heisenberg algebra (with $q$ a non-zero parameter), may be defined in terms of generators $a,a^\dagger,q^\frac{H}{2},q^{-\frac{H}{2}}$ and $1$ and relations:
$$
q^{\pm\frac{H}{2}}q^{\mp\frac{H}{2}}=1, \ \ \ \ \ [q^\frac{H}{2},a]=[q^\frac{H}{2},a^\dagger]=0, \ \ \ \ \ [a,a^\dagger]=\frac{q^H-q^{-H}}{q-q^{-1}}
$$
(Of course now $[.,.]$ is no more the Lie bracket but simply the usual commutator). This is known to be a quasitriangular hopf algebra. It may be thought of, as the $q$-deformation $U_q(L_H)$ of the universal enveloping algebra $U(L_H)$ of the Heisenberg Lie algebra $L_H$.
It can be shown, that, if $q$ is a real number, then the unitary representations of $U_q(L_H)$ are parameterized by a real, positive parameter $\hbar$. If we denote the basis vectors by $$H_\hbar=\{|n,\hbar\rangle\big{|}n=0,1,2,...\}$$
then the action of the generators is given by:
$$
|n,\hbar\rangle=\frac{(a^\dagger)^n}{[\hbar]^{\frac{n}{2}}\sqrt{n!}}|0,\hbar\rangle, \ \ \ \ \ q^{\pm\frac{H}{2}}|0,\hbar\rangle=q^{\pm\frac{\hbar}{2}}|0,\hbar\rangle, \ \ \ \ \ a|0,\hbar\rangle=0
$$
where $[\hbar]=\frac{q^\hbar-q^{-\hbar}}{q-q^{-1}}$. This a deformation of the usual Fock representation of the Heisenberg Lie algebra. If you are interested in similar examples, you can find more in S. Majid's book, "Foundations of Quantum group theory".
Furthermore, various $q$-deformations of the harmonic oscillator algebra can be used for a more systematic way of constructing such examples: Although most of $q$-deformed CCR are not quantum groups themselves (up to my knowledge there are generally no known hopf algebra structures for such algebras), suitable homomorphisms from quantum groups $U_q(g)$ (with $g$ being any Lie (super)algebra) to $q$-deformations of the harmonic oscillator can be used (such homomorphisms are usually called "realizations" in the literature) to pull back the $q$-deformed fock spaces of the $q$-deformed oscillators to representations of the corresponding quantum group $U_q(g)$.
Such methods have been applied since the '80's: L. C. Biedenharn and A. J. Macfarlane have provided descriptions of $su_q(2)$ deformations and their corresponding representations. A more complete account can be found in: Quantum Group Symmetry and Q-tensor Algebras. Similar methods have been used for the study of $su_q(1,1)$ deformed lie algebra representations. Two parameter $(q,s)$-deformations have also been studied, see for example the case of $sl_{q,s}(2)$ ... etc. These methods have also been extended to the case of deformations of the universal enveloping algebras of Lie superalgebras as well. See for example this article or this one.
The mathematical physics literature abounds of such examples during the last couple of decades.
The situation is similar to the way, various bosonic or fermionic realizations of Lie (super)algebras have been used to construct Lie (super)algebra representations, initiating from the usual symmetric/antisymmetric bosonic/fermionic Fock spaces: Such are the Holstein-Primakoff, the Dyson or the Schwinger (see: ch.3.8) realizations (see also: Dictionary on Lie Algebras and Superalgebras).
If you are interested in such topics (in the deformed or the undeformed sense), i can provide further references (i have done some work on similar stuff during my phd thesis).
Best Answer
Edited in light of clarification made by OP in comments to his question:
Yes, the result you want is proved by Lam and Shimozono; it is Theorem 10.16 of their paper arXiv:0705.1386. Their theorem (which is followed by a proof) identifies a localization of $QH^T(G/P)$ with a localization of a quotient of the torus equivariant homology of the affine Grassmannian; specialization gives the earlier non-equivariant unpublished result of Peterson.
The Lam/Shimozono result depends on an earlier calculation (in arXiv:math/0501213) by Mihalcea of the equivariant quantum product of a Schubert class by a divisor class; this rule should already suffice to determine the QDE.
arxiv:1007.1683 by Leung and Li is the state of the art in relations between $QH(G/P)$ and $QH(G/B)$, as far as I am aware. See in particular Theorem 1.4 (which however restricts to the case P/B equal to a flag variety).