In "commutative geometry," I think bimodules tend to be a little concealed. People are more likely to talk about "correspondences" which are the space version of bimodules: A correspondence between spaces X and Y is a space Z with maps to X and Y.
When you think in this language, there are lots of examples you're missing. For example, the right notion of a morphism between two symplectic manifolds is a Lagrangian subvariety of their product, or even a manifold mapping to their product with Lagrangian image (maybe not embedded). See, for example, Wehrheim and Woodward's Functoriality for Lagrangian correspondences in Floer homology.
Similarly, correspondences are incredibly important in geometric representation theory. See, for example, the work of Nakajima on quiver varieties.
The theory of stacks also is at least partially founded on taking correspondences seriously as objects, and in particular being able to quotients by any (flat) correspondence.
This same philosophy also underlies groupoidification as studied by the Baez school (they tend to use the word "span" instead of "correspondence" but it's the same thing).
Of the topics you mentioned, perhaps Representation Theory (of Lie (super)algebras) has been the most useful. I realise that this is not the point of your question, but some people may not be aware of the extent of its pervasiveness. Towards the bottom of the answer I mention also the use of representation theory of vertex algebras in condensed matter physics.
The representation theory of the Poincaré group (work of Wigner and Bargmann) underpins relativistic quantum field theory, which is the current formulation for elementary particle theories like the ones our experimental friends test at the LHC.
The quark model, which explains the observed spectrum of baryons and mesons, is essentially an application of the representation theory of SU(3). This resulted in the Nobel to Murray Gell-Mann.
The standard model of particle physics, for which Nobel prizes have also been awarded, is also heavily based on representation theory. In fact, there is a very influential Physics Report by Slansky called Group theory for unified model building, which for years was the representation theory bible for particle physicists.
More generally, many of the more speculative grand unified theories are based on fitting the observed spectrum in unitary irreps of simple Lie algebras, such as $\mathfrak{so}(10)$ or $\mathfrak{su}(5)$. Not to mention the supersymmetric theories like the minimal supersymmetric standard model.
Algebraic Geometry plays a huge rôle in String Theory: not just in the more formal aspects of the theory (understanding D-branes in terms of derived categories, stability conditions,...) but also in the attempts to find phenomenologically realistic compactifications. See, for example, this paper and others by various subsets of the same authors.
Perturbative string theory is essentially a two-dimensional (super)conformal field theory and such theories are largely governed by the representation theory of infinite-dimensional Lie (super)algebras or, more generally, vertex operator algebras. You might not think of this as "real", but in fact two-dimensional conformal field theory describes many statistical mechanical systems at criticality, some of which can be measured in the lab.
In fact, the first (and only?) manifestation of supersymmetry in Nature is the Josephson junction at criticality, which is described by a superconformal field theory. (By the way, the "super" in "superconductivity" and the one in "supersymmetry" are not the same!)
Best Answer
I think it is helpful to remember that there are basic differences between the commutative and non-commutative settings, which can't be eliminated just by technical devices.
At a basic level, commuting operators on a finite-dimensional vector space can be simultaneously diagonalized [added: technically, I should say upper-triangularized, but not let me not worry about this distinction here], but this is not true of non-commuting operators. This already suggests that one can't in any naive way define the spectrum of a non-commutative ring. (Remember that all rings are morally rings of operators, and that the spectrum of a commutative ring has the same meaning as the [added: simultaneous] spectrum of a collection of commuting operators.)
At a higher level, suppose that $M$ and $N$ are finitely generated modules over a commutative ring $A$ such that $M\otimes_A N = 0$, then $Tor_i^A(M,N) = 0$ for all $i$. If $A$ is non-commutative, this is no longer true in general. This reflects the fact that $M$ and $N$ no longer have well-defined supports on some concrete spectrum of $A$. This is why localization is not possible (at least in any naive sense) in general in the non-commutative setting. It is the same phenomenon as the uncertainty principle in quantum mechanics, and manifests itself in the same way: objects cannot be localized at points in the non-commutative setting.
These are genuine complexities that have to be confronted in any study of non-commutative geometry. They are the same ones faced by beginning students when they first discover that in general matrices don't commute. I would say that they are real, fascinating, and difficult, and people have put, and are currently putting, a lot of effort into understanding them. But it is a far cry from just generalizing the statements in Hartshorne.