Even if you assume any frame is valid, you have to pick inertial frames. So even if you travel few days, you will be off from earths orbit into space since earth is in circular motion which is acceletared

For a geometric center you would need a boundary of the universe

Tine machine probably moved in its own inertial reference frame. That will actually get you lost in space because the inertial frame does not orbit around, which involves rotation(rotation is intrinsically non-inertial, i.e accelerating). Time machine’s frame will be moving in a straight line if its inertial

Earth frame isn’t inertial

That’s what einstein said. There is no fixed reference frame, but only relative ones. Every “inertial”(meaning, motion without any external force) frame of reference is equally valid as any other inertial frame movibg with respect to it.

But for sure we can tell earth’s orbit is not inertial since circular motion occur, which is due to external force of gravity.

Edit:typo

It may use less aluminum, as the top and bottom surfaces are much thicker