Is it a postulate that light travels on a geodesics? In arbitrary spacetime, light travels on geodesics. Is this a postulate or can it be derived from a more fundamental principle?
 A: If one starts out by asserting the Einstein field equation, then one can deduce from that that a particle in freefall with non-zero rest mass follows a timelike geodesic. The method is based on the observation that $\nabla_a g_{bd} = 0$, combined with the field equation, leads to $\nabla_\mu T^{\mu b} = 0$. I guess (but have not proved it myself) that by treating a null geodesic as a limiting case of suitable timelike ones, the proof could be extended to a null geodesic for a particle with zero rest mass. ... but it is debatable whether the field equation could be said to be "more fundamental" (however that is to be defined) than the notion that freefall motion is geodesic.
A: The fact that particles move along geodesics isn't really an extra postulate. It essentially hinges on the Bianchi identity. Einstein's equations are
$$
G^{\mu\nu} = 8 \pi G T^{\mu \nu}.
$$
Using the Bianchi identity you can derive that
$$
\nabla_\mu G^{\mu \nu} = 0
$$
which implies
$$
\nabla_{\mu} T^{\mu \nu} = 0.
$$
If your stress energy tensor is described by a sort of "bump like" pulse of light, this equation means that the light pulse will follow a geodesic in spacetime. Of course, you also have to assume that the bump doesn't have so much energy that it deforms the spacetime around itself appreciably.
A: When you refer to light as travelling on geodesics, you are describing it as a ray, and thus, modelling it by geometrical optics. Geometrical optics, which can be derived from wave propagation, which in turn follows from Maxwell's equations, concludes that light takes the shortest path, or a geodesic.
In a similar way, the Lagrangian of general relativity combined with the Lagrangian of the electromagnetic field could be simplified to the analogous geometrical optics of general relativity, which must result in geometric light beams taking light-like 4D geodesics. Don't ask me about the math, but there is no doubt that you don't need to add that to the theory artificially.
A: We can call certain things axioms and others theorems, but that depends on what we choose as our set of axioms. For example, the Pythagorean theorem is a theorem if we take Euclid's postulates as our axioms, but in a Cartesian approach the Pythagoran "theorem" could actually be just an axiom, our definition of distance.
The fact that light travels on a geodesic is certainly hard to avoid because of its logical ties to other facts.
For example, suppose that a ray of light initially started along some timelike geodesic but then started following a null geodesic, so that when we splice together these two pieces, we don't have something that is a geodesic over all. This is unacceptable for a variety of reasons. It would automatically violate conservation of energy-momentum. It would also violate rotational symmetry -- for suppose an observer is at rest relative to the ray when the ray's motion is timelike. Then that observer says that the ray, which was initially at rest, spontaneously violated rotational symmetry by picking some random direction to head off in. (Here we implicitly assume that a timelike geodesic is a possible motion of an observer. We could take that as an axiom, but it's a matter of taste.)
If a ray starts on some null geodesic and then switches to some other null geodesic, then we get similar issues. The difference between the initial and final energy-momentum vectors is spacelike, so symmetry has been broken by picking the direction of this spacelike vector.
Note that bodies need not follow geodesics in free-fall motion if they are large or massive, or if energy conditions are violated. For example, if a sphere is spinning, then its spin axis picks out a preferred direction, so the symmetry arguments above will fail.
A: You're assuming that physics is an axiomatic discipline. It's not. Although, Hilbert stated that the axiomatic development of physics is an outstanding problem in the early 20C, and whilst there has been ongoing work since then (and before), it's nevertheless the case that physics hasn't been reduced to axiomatics. Hence, there is no more 'fundamental' postulate.
If you look at the history of physics, particularly relativity, the main reason to adopt the constancy and invariance of the speed of light as a fundamental principle is that it helped unify classical mechanics and classical electromagnetism.
