Answer 1:
The nature of time is a subject of physics that is very difficult to understand, and in many cases will lead to answers that are not intuitive at all based on our every day experiences! In fact, let me admit right now that this is not a subject that I fully understand, but I'll do my best to (hopefully) get you thinking about things in new ways, and possibly stimulate your curiosity enough to further research this topic.
First, I believe that the concept of "time" may be a little more complicated than simply the "relative position of two things as they are moving around." If you have ever watched Star Trek or some other SciFi type of show, you've probably heard the term "spacetime continuum." Space can be thought of as the three dimensional representation of everything around us. Time, in a sense, can be thought of as a fourth dimension. As it turns out, space and time cannot be clearly separated. Much like you cannot describe any physical object around you in two dimensions, you cannot describe events in our universe without incorporating time. Thus anything that happens in the universe is an event in both space and time, or the spacetime contiuum.
Now, you asked  what happens to time if things stopped moving? The answer to this is not simple, and in fact, is related Albert Einstein's Special Theory of Relativity. Let's begin by first thinking about the question by asking, "if things stopped moving with respect to what?" If you sit perfectly still in a chair, would have any kinetic energy? Basically not, if your frame of reference is say the room you're in. But we know that the earth is spinning, so you are actually moving through space. Further, the spinning earth is orbitting the sun, which is in an arm of a spiral galaxy (the Milky Way) that is rotating, and that galaxy is moving through space...well, you get the idea. From this, we realize that there is not an "absolute" frame of reference in the universe  everything is in motion relative to everything else.
So now here's the tricky part. Let's expand and slightly generalize your question to: Is time "absolute"  is time the same everywhere in the universe? This is what Einstein's Special Theory of Relativity addresses. Basically, the answer is no  time is not absolute! Time changes depending on your frame of reference. Without going in details, let's just skip to the "answer" from a simplistic point of view  time runs more slowly for a moving object than for a stationary object. One could argue this isn't "technically" true, but it's alright when first thinking about this. For example, let's imagine two clocks perfectly synchronized to each other. Now let's put one clock on a rocket and fly it to the moon and back. When the plane lands and we compare the clocks, we would find (if time is measured accurately enough) that the clock that was on the plane will be just behind the clock that we had kept stationary! We do not observe these changes (or things in our every day life) because they are not moving fast enough. These "time dilations" really only become observable as object approach the speed of light. You can do the calculation yourself  the relativistic effects are described by: [1(v/c)^2]^(1/2), where v is velocity, and c is the speed of light, or about 186,000 miles per second. A value of 1 is "normal" and deviations from 1 describe the dilations. Put in some numbers and see how fast things have to go to noticeably see deviations from 1.
There is an interesting thought experiment regarding the Special Theory of Relativity known as the "Twins paradox" that I encourage to research and think about.
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