Two Sentence Horror

This notion has always bothered me, so I thought a lot about it just now: A fundamental principle of Einstein's theory of Special Relativity is there is no privileged reference frame. That is, all measures of an object's motion are relative to other objects. There is no same-spot-in-space-but-5-minutes-ago for all observers. That's going to vary by reference frame.

See, the time machine before the trip does not move relative to Earth, but does move relative to the Sun, to the other planets, to the galactic core, et cetera. If we stipulate that going back 5 minutes equals a certain ∆x, that establishes the Earth's reference frame as privileged, and then observers in those other reference frames (given ∆t = -5 minutes) would calculate different values for c.

The way to avoid breaking physics is to recognize that observers in other reference frames could see what the distance that a trip of 5 minutes back in time means in Earth's inertial reference frame, but they'd calculate a different ∆t. But then, that means that time is just like space— all relative. How would a machine gain traction on it, what with no guideposts or reference points, no coordinate system with which to locate a target time? How would it measure out 5 minutes, except by internal chronometer? Then, going back in time 5 minutes would take 5 minutes for the occupant of the machine. (In this case, the machine going backwards in time would physically collide with itself-prior-to-the-trip in the first moments. Whoops.)

Assuming some magic technology that lets a time machine calculate some target time and pop out of existence now and pop into existence then, like in most sci-fi, it would have to calculate a destination location at which to pop into existence, too. Space and time are inextricably linked after all. (The Earth is a pretty big local mass, it should be easy to detect with... magic.) By coincidence, the time machine would also be a teleporter, and known physics would just be bits of rubble on the floor.

Anyway, other details that bother me: How does the time travel mechanism discriminate between machine and non-machine matter? Does it take a bubble of soil and air with it in order to be safe? Does the travel process necessarily sap the machine's momentum, or at least increase its mass? If not, it'll keep moving at the same velocity before and after the trip, and the Earth will never catch up.