Earth science relative dating practice

So you, I, and a group of C-14 atoms in a fossil will all experience the same time dilation. If there were rotational time dilation, and orbital time dilation, wouldn't there also be galactic or universal time dilation according to the speed with which our elements are moving through the larger space. Also, isn't it known that those speeds are changing fairly rapidly as the universe expands?

Third: We would experience the same time dilation, true, but not if we weren't in the same time frame.

Theoretically if in early years the earth was moving at an astronomically different pace (whether in orbit or rotation, whatever sounds better) wouldn't that alter the science of carbon dating?

As I understand time relativity, If the speed at which the earth was moving changes, the elements and organisms there-in would experience time at a different rate.

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