No, it has become trivial! Once you know it's a wave equation everything follows very simply! All the results about the bottom, gravity, etc. Had it been a diffusion equation, or elliptic, then the bottom would have fallen (out of this brain teaser) immediately!

But this particular scenario is NOT governed by the wave equation because it's a slinky, not a general spring with an open-coil rest state. The wave equation involves only simple elasticity, not the highly non-linear effects of the slinky coils collapsing to a solid cylinder.

Thus, this it is not a simple kx vs. mg in that k has extremely low value for some parts of the system and at some times but then becomes vastly greater when the coils collapse and contact each other.

It cannot be anything else because the only forces in the system are the spring tension T and gravity. Slinky is a bit pretentious (Tp - is the pretension force what you meant was missing?), so stretching it creates an opposing tension, T = Tp - kx. When you hold the slinky by the topmost coil, it stretches out until the force of gravity is balanced by the tension. When you release the top, the consecutive coils start to collapse onto each other from the top as the "deformation" propagates through the spring in a wave-like manner: [$]\partial^2 \psi/\partial t^2 = T_p/\rho\ \partial^2 \psi/\partial x^2[$]. I think the beauty of the experiment is in how the centre of mass of the slinky experiences a free fall, while the slinky as a whole is in zero-g. It's the "opposite" situation to popular experiments demonstrating free fall with water in a punched bottle - when it falls, the water doesn't leak, because it's in zero-g.

It depends on what we mean by "the" wave equation. In terms of the basic set-up, sure it's all about an interaction of kx and g in the distributed mass of the spring. But there's two values of k in the system, the very very low uncompressed one that creates waves traveling on the order of a few meters a second and a very very high value of k for the compressed spring that creates waves traveling within the compressed section on the order of a several thousand meters a second.

There are combinations of g, k, L, and Tp that create springs that don't fully collapse when dropped in this manner. They'd show different dynamics with a more modest compression wave traveling down and up the spring as it fell.