To elaborate on Kevin's excellent answer, one can account for the current absence of "higher loop" representation theory using physics. Namely, all of the representation theoretic structures you mention fit in very naturally into the study of gauge theory, specifically 4-dimensional $\mathcal N=2$ gauge theories. These come in two main classes (with some intersection) - the quiver gauge theories, which are the natural homes for algebras like Yangians, quantum loop algebras, and elliptic quantum groups; and the class S theories (reductions of the 6d "theory $\mathfrak X$" -- the (2,0) superconformal field theory labeled by a Dynikin diagram - on Riemann surfaces), which are the natural home for geometric Langlands, double affine Hecke algebras, Khovanov homology etc. (the theory Kevin describes associated to $U_q(\mathfrak g)$ is $\mathcal N=4$ super Yang Mills, which is the case when the Riemann surface is the two-torus).
So why should this be relevant? the question of attaching interesting representation theory to maps into Lie groups is very closely linked to the question of finding interesting gauge theories in higher dimensions (the latter is strictly stronger but seems like the most natural framework we have for such questions). Specifically, we want supersymmetric gauge theories, if we want them to have any relation to topological field theory or algebraic geometry etc.
However there are no-go theorems for finding gauge theories in higher dimensions. Even at the classical level it is impossible (thansk to Lie theory, namely the structure of spin representations) to have a supersymmetric gauge theory in more than 10 dimensions ---- any SUSY theory in dimensions above ten also includes fields of spin two and above (so physically is a theory of gravity), while above dimension 11 we have to have higher spin fields still (which physicists tell us doesn't make sense -- regardless it won't be a gauge theory). In any case theories with gravity and other stuff are a very far stretch to be called representation theories!
At the quantum level (which is what we need for representation theory) it's much harder still -- I believe there are no UV complete quantum gauge theories above dimension 4 (in other words higher dimensional theories have to have "other nonperturbative stuff in them").
All of the representation theoretic structures you mention naturally fit into theories that come from six dimensions at best (reduced to 4 dimensions along a plane, cylinder or torus in the quiver gauge theory case to see Yangians, quantum affine algebras and elliptic quantum groups, or along a Riemann surface in the class S case). Studying in particular theory $\mathfrak X$ on various reductions gives a huge amount of structure, and includes things like ``triple affine Hecke algebras" presumably when reduced on a three-torus, but there's a clear upper bound to the complexity you'll get from these considerations.
Now of course you might ask what about theories that don't come from supersymmetric gauge theory? the only interesting source I've heard of for higher dimensional topological field theories is (as you hint) chromatic homotopy theory, in particular the fascianting work of Hopkins and Lurie on ambidexterity in the$K(n)$-local category. This is a natural place to look for "higher representation theory", which is touched on I believe in lectures of Lurie -- but my naive impression is these theories will have a very different flavor than the representation theory you refer to (in particular a fixed prime will be involved, and these theories certainly don't feel like traditional quantum field theory!). But it's a fascinating future direction. For a hint of what kind of representation theory this leads to we have the theorem of Hopkins-Kuhn-Ravenel describing the $n$-th Morava K-theory of BG in terms of n-tuples of commuting elements in G --- i.e. the kind of characters you might expect for G-actions on $(n-1)$-categories.
Here is the picture, as I understand it, for $\mathrm{GL}_n$; this is described in chapter 8 of Godement-Jacquet (and see also the archimedean theory in Jacquet-Langlands).
Let $F \in \{\mathbb{R},\mathbb{C}\}$ be an archimedean local field. Let $\mathcal{H}_1$ denote the space of smooth compactly supported functions on $\mathrm{GL}_n(F)$ that are bi-$K$-finite, where $K = \mathrm{U}(n)$ if $F = \mathbb{C}$ and $K = \mathrm{O}(n)$ if $F = \mathbb{R}$. These may be regarded as measures on $\mathrm{GL}_n(F)$, in which case $\mathcal{H}_1$ is an algebra under convolution: for $f_1, f_2 \in \mathcal{H}_1$,
\[f_1 \ast f_2(g) = \int_{\mathrm{GL}_n(F)} f_1(gh^{-1}) f_2(h) \, dh.\]
Every function $\xi$ on $K$ that is a finite sum of matrix coefficients of irreducible representations $\tau$ of $K$ may be identified with a measure on $K$, and hence on $\mathrm{GL}_n(F)$. Under convolution, these measures form an algebra $\mathcal{H}_2$. We let $\mathcal{H}_F = \mathcal{H}_1 \oplus \mathcal{H}_2$. This is an algebra under convolution of measures: for $f \in \mathcal{H}_1$ and $\xi \in \mathcal{H}_2$,
\[\xi \ast f(g) = \int_{K} \xi(k) f(k^{-1} g) \, dk\]
and
\[f \ast \xi(g) = \int_{K} f(gk^{-1}) \xi(k) \, dk.\]
This is the Hecke algebra of $\mathrm{GL}_n(F)$. Given a representation $(\pi,V)$ of $\mathrm{GL}_n(F)$, we define the action of $f \in \mathcal{H}_F$ on $v \in V$ by
\[\pi(f) \cdot v = \int_{\mathrm{GL}_n(F)} f(g) \pi(g) \cdot v \, dg.\]
Best Answer
The early work of Borel showed in effect how to interpret the cohomology algebra of a flag variety as the coinvariant algebra associated to the Weyl group, which affords the regular representation of $W$.(A useful exposition, if you can locate it, is the 1982 Pitman Research Notes Geometry of Coxeter Groups by Howard Hiller, though the account of Coxeter groups in general is too sketchy.) This has been understood from various viewpoints such as compact groups and complex semisimple Lie groups, with further combinatorial development by Bernstein-Gelfand-Gelfand and Demazure. By now there is a lot of related literature, including connections with the Springer Correspondence.
On the other hand, the Iwahori-Hecke algebra associated to the Coxeter group $W$ is a deformation of the integral group ring, therefore also close to the group algebra and representation theory of $W$. A fundamental source is the 1979 Invent. Math. paper by Kazhdan and Lusztig here. But as far as I know the answer to the original question (how is the cohomology ring related to the Hecke algebra) is that the two are only indirectly related.
P.S. Bischof has pointed to some of the rich developments following the papers by BGG and Demazure, but I'm not enough of a specialist to know which references are most relevant here. One important direction I should mention involves work by Kostant and Kumar on what they call the "nil Hecke ring", motivated partly by wanting to work with analogues of Schubert varieties in the infinite dimensional Kac-Moody framework. See in particular: Bertram Kostant and Shrawan Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Adv. in Math. 62 (1986), no. 3, 187–237.