Is time on earth, relative to everywhere else, dependent on the earth's speed? Earth rotates at a speed, it moves around the sun, the sun moves around the galaxy and the galaxy is also moving - is it that cumulative speed that give us our version of time? Setting aside the catastrophe of speeding up or slowing down the earth in anyone of these 'speeds', would a change in any one of them impact time relative to an observer not on earth? How fast would a rocket have to go in the opposite direction to stop moving relative to earth, the galaxy etc. and would time then stop for anything/one in that capsule?
Short answer: It depends...
Long answer: ...on what you're measuring. Take your example about the earth moving and the sun moving through the moving galaxy. We can take our frame of reference to be, say, the Andromeda galaxy. The effect of time dilation differs between these two points. So by 'our version of time', it's more like a change in time relative to some other place; it's not an absolute value.
Think of it like this. Say you wanted to measure speed in the most direct way; measure the distance between point a and point b, and measure the rate at which it changes. The difference in time between those two points is related to that relative velocity. If an object c was orbiting a, then you would have to pick; am I measuring the speed relative to a or b. And when you're measuring time dilation (with special relativity), it is the difference between the time experienced between those to points.
While it may not be strictly true, I find it can be useful to think of time dilation as some kind of Δx, much like how distance is Δd (the change in position between two points).