[Oberon]

It all comes down to the nature of space. Is it a vacuum, a fluid or what else? If there are effects produced by motion relative to absolute space then yes. If one could measure the one way speed of light in every direction, this may show absolute motion through non congruent light paths (very similar to the Sagnac effect, except in a linear translation).

What would the implications be, then, for a relativistic understanding of the universe? Would Lorentz calculations require adjustment based on absolute velocity, then?

[Gary Boldwater]

Lorentz contractions would be relative to the preferred reference frame. Search on "quantum foam" and "gas filled Michelson interferometer" for some fun reading.

Some things that are not relative are acceleration and radiation. If one accelerates charge, it radiates (not in every case). If I accelerate relative to stationary charge I don't see the radiation.