James
You ask what happens physically to cars and drivers on rapid acceleration (frame change). I confirm I derive a Doppler shift, which is a physical length contraction, or dilation, if the body is compressible. So the bodies DO "suffer" if they themselves accelerate. It is purely an unfamiliar way of re-appraising the very familiar. The experimental evidence fills many a scrap yard and hospital. More gentle frame changes are recommended, and smaller masses do less damage to larger ones (some old Law I recall!).
And when we consider very small masses (or even zero if you wish) like photons, the same happens because they are compressible, as is a sequence of waves. The windscreen of the car hardly notices the impact (the scattering may even be 'non-elastic' for the scatterer), but the photons or waves compress (blue shift).
If they're coming in by the rear screen they will be wider apart after detection (which should be though of as a 'sampling' process) as the car has moved a bit between each one arriving (again either waves or a string of individual 'particles' as you wish).
So thinking 'photon'; the effect is that wavelength and frequency change inversely on negotiating the ('fluid dynamic' coupling zone) so c changes to c'. This produces the LOCAL REALITY Einstein sought all his life, by producing the SR postulates from a quantum mechanism, giving the unification Einstein sought all his life.
The problem is that this is all so unfamiliar (as Feynman said it would be) and simple (as Wheeler said it would be) and conceptual (as Einstein said it would be) that it is largely unrecognizable to most formally indoctrained physicists uless the necessary assumptions identified are suspended to test it.
Lastly clocks moving at the surface. If the observer is moving with them, No, they won't change. Think about what they are 'moving' with respect to. If they do our rotational velocity they are at rest in the ECI frame!! How do they know if they are moving or not! The concept 'Moving' always has to be specified wrt the 'LOCAL BACKGROUND.' Only if a clock is in another frame does 'Proper Time' not apply, so flight time has to be allowed for, and relative change of position, so apparent change of clock speed. Simple but normally ignored.
Your last question; You must decide precisely what type of clock as mechanisms vary and are affected differently by many things. But sorry, No. Time itself would not change as it does not 'exist' in terms of an 'entity' existing. 'Emissions rates' change for many reasons, and apparent emission rates change also due to observer motion. That would be all to be consistent with the rest of the ontology.
I hope that position is clear, but it is initially apparently complex, so if not do ask again.
best wishes
Peter