It is often claimed that light follows the shortest path, but that is not necessarily true. Under the right conditions, reflecting from inside an ellipsoidal mirror, it will instead follow the longest path, something you can readily test with a polished spoon. A statement closer to reality is that it follows an extreme path; a shortest or longest.
And even that doesn’t get to the heart of it. Any photon follows all paths but most paths interfere destructively with each other since since very near paths differ in length by a half wavelength. This fails to happen only at extreme points. This effect can be demonstrated by using grating ruled so that, at a test wavelength, the half wavelength shorter and longer paths are destroyed.
Those aren’t the longest paths. There is no “longest path”; for any given path you can always construct a really wild fractal-ish path which is longer. Even if there’s a shorter path, photons will always follow the locally-shortest path. That is to say, any decrease in the path length would require you to change the path by a large amount.
And when something emits light, mass is still constant, as long as you consider the light travelling off to be part of the system. As I said before, even though an individual photon does not have mass, a system of them in general does. However, a system of photons will typically have higher entropy than other systems with the same total mass, so things radiating photons do, in general, increase the entropy of the Universe. And in the heat death of the Universe, most of the mass will in fact be in photons and other light particles.