Is the Born rule a fundamental postulate of quantum mechanics? Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution? 
 A: The use of the word "postulate" in the question may indicate an unexamined assumption that we must or should discuss this sort of thing using an imitation of the axiomatic approach to mathematics -- a style of physics that can be done well or badly and that dates back to the faux-Euclidean presentation of the Principia.  If we make that choice, then in my opinion Luboš Motl's comment says all that needs to be said. (Gleason's theorem and quantum Bayesianism (Caves 2001) might also be worth looking at.) However, the pseudo-axiomatic approach has limitations. For one thing, it's almost always too unwieldy to be usable for more than toy theories. (One of the only exceptions I know of is Fleuriot 2001.) Also, although mathematicians are happy to work with undefined primitive terms (as in Hilbert's saying about tables, chairs, and beer mugs), in physics, terms like "force" or "measurement" can have preexisting informal or operational definitions, so treating them as primitive notions can in fact be a kind of intellectual sloppiness that's masked by the superficial appearance of mathematical rigor.
So what can physical arguments say about the Born rule?
The Born rule refers to measurements and probability, both of which may be impossible to define rigorously. But our notion of probability always involves normalization. This suggests that we should only expect the Born rule to apply in the context of nonrelativistic quantum mechanics, where there is no particle annihilation or creation. Sure enough, the Schrödinger equation, which is nonrelativistic, conserves probability as defined by the Born rule, but the Klein-Gordon equation, which is relativistic, doesn't.
This also gives one justification for why the Born rule can't involve some other even power of the wavefunction -- probability wouldn't be conserved by the Schrödinger equation. Aaronson 2004 gives some other examples of things that go wrong if you try to change the Born rule by using an exponent other than 2.
The OP asks whether the Born rule follows from unitarity. It doesn't, since unitarity holds for both the Schrödinger equation and the Klein-Gordon equation, but the Born rule is valid only for the former.
Although photons are inherently relativistic, there are many situations, such as two-source interference, in which there is no photon creation or annihilation, and in such a situation we also expect to have normalized probabilities and to be able to use "particle talk" (Halvorson 2001). This is nice because for photons, unlike electrons, we have a classical field theory to compare with, so we can invoke the correspondence principle. For two-source interference, clearly the only way to recover the classical limit at large particle numbers is if the square of the "wavefunction" ($\mathbf{E}$ and $\mathbf{B}$ fields) is proportional to probability. (There is a huge literature on this topic of the photon "wavefunction". See Birula 2005 for a review. My only point here is to give a physical plausibility argument. Basically, the most naive version of this approach works fine if the wave is monochromatic and if your detector intercepts a part of the wave that's small enough to look like a plane wave.) Since the Born rule has to hold for the electromagnetic "wavefunction," and electromagnetic waves can interact with matter, it clearly has to hold for material particles as well, or else we wouldn't have a consistent notion of the probability that a photon "is" in a certain place and the probability that the photon would be detected in that place by a material detector.
The Born rule says that probability doesn't depend on the phase of an electron's complex wavefunction $\Psi$. We could ask why the Born rule couldn't depend on some real-valued function such as $\operatorname{\arg} \Psi$ or $\mathfrak{Re} \Psi$. There is a good physical reason for this. There is an uncertainty relation between phase $\phi$ and particle number $n$ (Carruthers 1968). For fermions, the uncertainty in $n$ in a given state is always small, so the uncertainty in phase is very large. This means that the phase of the electron wavefunction can't be observable (Peierls 1979).
I've seen the view expressed that the many-worlds interpretation (MWI) is unable to explain the Born rule, and that this is a problem for MWI. I disagree, since none of the arguments above depended in any way on the choice of an interpretation of quantum mechanics. In the Copenhagen interpretation (CI), the Born rule typically appears as a postulate, which refers to the undefined primitive notion of "measurement;" I don't consider this an explanation. We often visualize the MWI in terms of a bifurcation of the universe at the moment when a "measurement" takes place, but this discontinuity is really just a cartoon picture of the smooth process by which quantum-mechanical correlations spread out into the universe. In general, interpretations of quantum mechanics are explanations of the psychological experience of doing quantum-mechanical experiments. Since they're psychological explanations, not physical ones, we shouldn't expect them to explain a physical fact like the Born rule.
Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," http://arxiv.org/abs/quant-ph/0401062
Bialynicki-Birula, "Photon wave function", 2005, http://arxiv.org/abs/quant-ph/0508202
Carruthers and Nieto, "Phase and Angle Variables in Quantum Mechanics", Rev Mod Phys 40 (1968) 411; copy available at http://www.scribd.com/doc/147614679/Phase-and-Angle-Variables-in-Quantum-Mechanics (may be illegal, or may fall under fair use, depending on your interpretation of your country's laws)
Caves, Fuchs, and Schack, "Quantum probabilities as Bayesian probabilities", 2001, http://arxiv.org/abs/quant-ph/0106133; see also Scientific American, June 2013
Fleuriot, A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia, Springer, 2001
Halvorson and Clifton, "No place for particles in relativistic quantum theories?", 2001, http://philsci-archive.pitt.edu/195/
Peierls, Surprises in Theoretical Physics, section 1.3
A: This is clearly a somewhat controversial topic, but Zurek claimed to derive the Born Rule from other postulates.
A: The idea of deriving the Born rule (and in fact the whole measurement postulate) from the usual unitary evolution of quantum systems is at the very heart of a realist interpretation of quantum theory. If the quantum state really describes a the true internal state of a system and measurement is just a certain kind of interaction, then there should be only one single law for the time evolution. 
Quantum theory however is fundamentally non-local and separating systems is conceptually hard, which makes observer and experiment impossible to describe separately. There should be a system containing both parts however and which follows a simple law of time evolution. Of course, the obvious candidate for such a law is unitary evolution, simply because that is what we observe for systems that we isolate as good as possible.
It is usually argued that this route leads to the Everett interpretation of quantum theory, where observations are relative to the observer and realized by entangled states. There have been several attempts to derive the Born rule in this context, but all that seem valid require additional assumptions that are questionable (and may in fact be inconsistent with the realist approach or other fundamental assumptions).
The reason why there cannot be a derivation that just uses ordinary unitary evolution and results in the Born rule is not even unitarity but the linearity of the theory. Say there is an evolution that takes out input to the measurement output, and we decide to measure a|A>+b|B> in the basis {|A>,|B>}. Then independently from the environment the Born rule predicts that |A> and |B> are invariant under measurement. A superposition (|A>+B>)/sqrt(2) should end up in either |A> or |B> depending on a possible environment state if the Born rule applies. The linearity of the theory requires that the outcome is a superposition of |A> and |B> however (the phase may change though). 
Everett's answer to this problem is that the superposition comes out, but with the outcomes entangled with the observer seeing either outcome. But this creates two observers that are unaware of their own amplitude. Because of the linearity their future evolution is independent from the branch amplitude, and it's therefore hard to argue that any aspects of their perceived reality would depend on the branch amplitude.
Interestingly approaches to fix this issue, like the use of decision theory, advanced branch counting, etc, in some form introduce a nonlinear element to the theory. Be it a measure of branch amplitude, a cutoff amplitude or amplitude discretization, a stability rule (envariance or quantum darwinism). There are also approaches that don't hide the nonlinearity in additional assumptions that may collide with the linear evolution. Those are explicit nonlinear variations of the Schroedinger equation that can in fact produce an evolution that allows the Born rule to emerge. Of course, this is not something that most theorists embrace, simply because the linearity of quantum theory is such an attractive feature.
But there's one more approach that I personally favor. The nonlinearity could be only subjective to an observer, caused by incomplete knowledge about the universe. An observer, i.e. a local mechanism realized within quantum theory, can only gather information by interacting with his environment. Certain information however is inaccessible dynamically, hidden outside the observer's light cone or just not available for direct interaction. Considering this, it can be shown that reconstructing the best possible state description an observer can come up with must follow a dynamic law that is not unitary all the time, but also contains sudden state jumps with random outcomes driven by incoming priorly unknown information from the environment. It can be shown that a photon from the environment with entirely unknown polarization can cause a subjective state jump that corresponds exactly to the Born rule. This is of course a bold claim. But please see http://arxiv.org/abs/1205.0293 for a proper derivation and discussion of the details. If you you would like to look at a more gently introduction to the idea you can also read the (less complete but more intuitive) blog I've set up for this: http://aquantumoftheory.wordpress.com
A: Strictly speaking, the Born rule cannot be derived from unitary evolution, furthermore, in some sense the Born rule and unitary evolution are mutually contradictory, as, in general, a definite outcome of measurement is impossible under unitary evolution - no measurement is ever final, as unitary evolution cannot produce irreversibility or turn a pure state into a mixture. However, in some cases, the Born rule can be derived from unitary evolution as an approximate result - see, e.g., the following outstanding work: http://arxiv.org/abs/1107.2138 (accepted for publication in Physics Reports). The authors show (based on a rigorously solvable model of measurements) that irreversibility of measurement process can emerge in the same way as irreversibility in statistical physics - the recurrence times become very long, infinite for all practical purposes, when the apparatus contains a very large number of particles. However, for a finite number of particles there are some violations of the Born rule (see, e.g., the above-mentioned work, p. 115). 
A: The Born rule is a fundamental postulate of quantum mechanics and therefore it cannot be derived from other postulates --precisely your first link emphasizes this--.
In particular the Born rule cannot be derived from unitary evolution because the rule is not unitary
$$A \rightarrow B_1$$
$$A \rightarrow B_2$$
$$A \rightarrow B_3$$
$$A \rightarrow \cdots$$
The Born rule can be obtained from non-unitary evolutions.
A: It is independent, but it is not fundamental, as it applies only to highly idealized kinds of measurements. (Realistic measurements are governed by POVMs instead.) 
In fact, the role of Born's rule in quantum mechanics is marginal (after the standard introduction and the derivation of the notion of expectation).
It is hardly ever used for the analysis of real problems, except to shed light on problems in the foundations of quantum mechanics. 
A: Yes it can be derived from unitarity of quantum evolution. This was shown by Lesovik in his paper, Derivation of the Born rule from the unitarity of quantum evolution.
