As long as nobody knows what gravity actually is, and how it relates to the other realities of the Universe, one can speculate. Of course, it is well known since Kepler and Newton, that it is the force holding masses together. We even have equations to use gravity.

Einstein did not know what gravity is and how it relates to the other known energies. His pasting gravity on mass as curvature of space is unreliable, since he started his model with gravity. There is no cause – effect determination. To determine what gravity is and how it relates to the other energies, one cannot start with gravity as one of the starting theorems.

I did model the formation of the Universe as a crystallization process, converting the pre-BB energy into mass particles. Gravity is not in the theorems. The surprising result (!) of the modeling was that gravity is anti-energy. Following up the processes in the Universe, this energy is in the form of photons.

Ingo

If you go to arXiv.org and ask for papers with Aether in the title during the years 2011-13 you'll get already 44 answers. Thus there is activity.

A fairly recent review that summarizes many of the constraints of the theory is given here: http://arxiv.org/abs/0801.1547

There has also been some recent activity that has made even more stringent constraints on the theory. Most notably, there are new constraints from binary pulsar observations that force the parameters of Einstein-Aether theory to be fairly small, from this paper: http://arxiv.org/abs/1311.7144.

From an effective field theory point of view, where we imagine that the aether field is the result of Lorentz symmetry being spontaneously broken, the parameters in the aether action (c1, c2, c3, and c4) would be expected to be of order (M/M_Planck)^2, where M is the energy at which Lorentz symmetry is broken. So unless the scale of Lorentz symmetry breaking is near the Planck scale, we would expect the aether parameters to be quite small. If these parameters are small, the observable effects of such a field are greatly diminished.

In terms of whether the theory is well-posed, I would say the answer is pretty much yes. The initial value formulation of the theory is fine, and you can determine the necessary constraints that need to be satisfied for initial data. Like GR, this theory is diffeomorphism invariant, so as always you need to deal with gauge fixing and all that, but everything works out fine.

The theory does have some counterintuitive features however. For example, there are constraints coming from the absence of gravitational Cherenkov radiation, which say that if the wave modes in Einstein aether theory move slower than the speed of light, you get strong constraints on the aether parameters. Most people then take this as a requirement that these wave speeds should be faster than light. Note that there isn't really anything wrong with this, since your theory is explicitly violating Lorentz symmetry, so there is no reason to expect all waves to move slower than light.

Perhaps one open problem from a theoretical viewpoint is a proof of the positive energy theorem for it. This has been shown for a particular class of Einstein-aether theories that are closely related to the classical limit of Horava gravity: http://arxiv.org/abs/1108.1835. The proof is nontrivial because the aether stress tensor does not satisfy the dominant energy condition, which is needed for the positive energy theorem of GR.