What are the eigenvalues of the Lorentz matrix?

A Lorentz transformation lets you compute an object's properties in one inertial frame, given its properties in another inertial frame. Inertial frames, by definition, do not accelerate. An accelerating object is always instantaneously at rest in some inertial frame.

Whether such-and-such is a law of nature is an experimental question. We have no evidence that Lorentz invariance is broken, but people are looking. You might look at the participants in this conference to get an idea of the field.

The most general Lorentz matrix is a product of three rotations and three boosts. For pure rotations we may always choose our coordinate system so that the Lorentz matrix has the form $$\left(\array{ 1\\ &1\\ &&\cos\theta & -\sin\theta \\ &&\sin\theta & \cos\theta \\ }\right).$$ A timelike vector, or a vector along the rotation axis, has eigenvalue 1, since they are not affected by rotations. In the plane of rotation the eigenvectors are $(1,\pm i)$; all real vectors in the plane of rotation get rotated. The corresponding eigenvalues are $e^{±i\theta}$.
 
Similarly, we may always choose our axes so that a boost is written $$ \left(\array{ \gamma & -\gamma\beta \\ -\gamma\beta & \gamma \\&&1\\&&&1}\right) .$$ Some algebra shows that the non-unity eigenvalues of this matrix are $$ \gamma (1\pm\beta) = \sqrt\frac{1\pm\beta}{1\mp\beta} $$ which is, as you say, the relativistic Doppler shift between an observer at rest and an emitter in the boosted frame. You can verify by hand that the corresponding eigenvectors are $(1,\pm1,0,0)$. These are the light-like worldlines on a Minkowski diagram: the paths taken by photons which would be found later to have the associated Doppler shifts.
 
A boost in a random direction would have the same four eigenvalues: $\gamma(1\pm\beta)$ for light-like vectors parallel and antiparallel to the boost, and unity for vectors in the spacelike plane perpendicular to the boost.