How is the physical meaning of an irreducible representation justified?
Invariant states are not the only meaningful ones. Even in classical mechanics, a baseball traveling 90 mph toward my head is quite meaningful to me, even though it is of no consequence to my fellow mathematician a mile away.
The focus on invariant subspaces comes not from an assumption, but from the way physicists do their work. They want to predict behavior by making calculations. They want to find laws that are universal. They want equations and calculation rules that will be invariant under a change of observers.
Any particular calculation might require a choice of coordinates, but the rules must be invariant under that choice. Once we're talking about one particular baseball trajectory, that trajectory will look different in different coordinate systems; the rules governing baseball flight, however, must look the same in all equivalent coordinate systems.
The natural features of baseballs arise from the equivalence classes of trajectories of baseballs -- equivalence under the group action. Here, if we pretend the earth is flat, gravity is vertical, and air does not resist the baseball, the group is generated by translations and rotations of the plane. Any physically natural, intrinsic property of the baseball itself (such as its mass) or its trajectory (such as the speed of the baseball) must be invariant under the group action. If you don't know a priori what these properties will be, a good way to find them is to pass from individual instances (the baseball heading toward me at 90mph) to the equivalence class generated by individual instances under the group action (the set of all conceivable baseballs traveling at 90mph). Note that the equivalence class is invariant under the group action, and it is exactly this invariance that makes the equivalence class a useful object of the physicists' study.
More generally, if you are studying a physical system with symmetry, it's a good bet that the invariant objects will lead to physically relevant, important quantities. It's more a philosophy than an axiom, but it has worked for centuries.
I suppose an answer to your question, as simple as possible is this: You would probably be happier if not an irreducible representation, but rather a single function was declared the object of interest (indeed this function, upon normalization, would literally be the probability distribution at a given time of finding the particle there). However, since the group G acts on the phase space of the system and preserves the energy functional under study (because it's an electron orbiting a proton, the energy is a function of r alone), any function f you find will transform under rotations by the element g to the element g.f.
Now suppose you and a friend are both conducting the experiment simultaneously, and such that your points of view of the proton are related by a rotation g about the proton. When you observe a distribution f, your friend will simultaneously observe the distribution g.f, just by computing change of coordinates, not by any physical assumption. Now, add the physical assumption that you and your friend should be able to reconcile your results when you meet to discuss them, and you'll find that one cannot make a distinction between f and g.f. So you shouldn't study single distributions f, but rather orbits g.f of a given function. This is my attempt at answering your second question. It is not a mathematical but physical answer, since you asked a physical question.
To your first question, I would simply agree with José Figueroa-O'Farrill that the term elementary state doesn't mean so much, except that you can formally express any state as a linear combination of its "elementary" states, and that this is useful for computing various quantitites (like the energy attributed to the state, using the Casimir element, as she does later in the book).
I suppose yet another perspective on the first question, which is (slightly) more mathematical than my original answer is that you want to study functions on R^3/SO(3). Since SO(3) action on R^3 is extremely non-free, the naiive approach of just taking invariant functions, would yield the wrong thing. Since R^3 is contractible, homotopy theory would tell us that R^3/SO(3) is homotopy equivalent to "pt/SO(3)", which in homotopy world is precisely the category RepSO(3). Maybe someone with a better grasp of homotopy theory can flesh this out?
I think at a later point in the book than you referred to (correct me if my memory is incorrect), Singer explains that you want to seek states which minimize the energy, which is either defined or derived, depending on your conventions, to be a simple function of the eigenvalues of a certain differential operator, which in this case coincides with the casimir element EF + FE + H^2/2 in the enveloping algebra of sl_2(C), which is the complexification of the lie algebra so_3. So the physical problem of finding states with minimal energy is reduced to finding eigenvalues of a certain operator, which by the above discussion commutes with the SO(3) action. Since the operators in SO(3) commute with the Casimir, they preserve its eigenspaces, which means that f and g.f get assigned the same energy. So, the unique up to scaling "s" distribution has minimal energy, the three "d" orbits have the next highest energy, and so on. One could certainly ask why don't all the electrons just adopt the s orbital, but this is forbidden by an observational law (Pauli's exclusion principle) stating that distinct electrons can't have the same distribution, or more precisely that the span of the distributions of n electrons has dimension n. Well actually n/2 because of "spin", but nevermind for now.
By the way, at the risk of being accused of distracting with a pretty picture, I recommend
http://www.orbitals.com/orb/orbtable.htm
which depicts the distributions we're discussing by drawing the crests where they are maximal in length as complex numbers. It's amazing how they match with the grayscale observed data, and how complicated they can get for large numbers of electrons. Note that they are definitely NOT G-invariant! (although a cool feature you can see is that precisely one in each level is H-invariant, where H is the rotation through a fixed axis. This happens because you have odd dimensional SO(3) reps, and the element H in so_3 has weights -k, -k+2, ..., 0 , ... k-2, k on each such V, and so since H has a one-dimensional zero weight space, it must fix one of the vectors and this means one function should be invariant about the axis defining H).
I'm not sure that there is here a physical assumption, as much as a convenient way in which to talk about the physical system. Quantum mechanical representations are unitary and, albeit typically infinite-dimensional, they are also typically fully reducible. Hence in a way, if you understand the irreducible representations you can understand any representation.
For instance, Wigner argues this way in his 1939 paper on the unitary representations of the Poincaré group. This might address the second version of your question about elementary particles corresponding to irreducible representations of the Poincaré group.
I guess that what I am trying to say is that a state corresponding to a reducible representation $V\oplus W$ is physically indistinguishable from one made up of two states corresponding to $V$ and $W$, respectively. Hence nothing is gained by considering such states as "elementary".