OK, this is a two-part question:
Do we experience time at the rate we do because of the speed that our planet/solar system/galaxy is moving through the universe (or multiverse, if you prefer)?
Is it possible to be at a complete standstill in space? How would you be able to tell? If you were at a standstill, would time cease to exist for you?
I believe you are making a conceptual mistake, which is that you think there is some “real” time, and everything else is relative to that (faster, slower, same speed). The thing is, there is no “proper” time for the universe.
You might think, “What if I put myself in the exact center of the universe, such that everything was moving away from me in a uniform speed distribution?” Well, the thing is, every point in the universe is pretty much like that.
That’s hard to visualize, but most simple books on relativity ask you to imagine a bunch of dots on an expanding balloon, and also assume that they can only see along the surface of the balloon. (This corresponds to two-dimensional and three-dimensional space standing in for our three- and four-dimensional space.)
Well, the funny thing is, every point on that expanding balloon will have the impression that everybody else is moving away from it. There’s no “point of no motion” by which to evaluate the “real” speed of the other points.
I hope that clears things up.
No.
It depends on how you define complete standstill. All motion is relative. Every inertial frame is equally valid. The best you can say is that in some frame you are not moving. Time does not stop when there is no (relative) motion. There is no way to tell one frame from another. Well, not really. You can detect motion relative to the background radiation, but that is just one particular frame.
I believe you are better at this stuff than I am DrM but wouldn’t the answer to #1 be a Yes?
We have us here on earth with our clocks. In another galaxy far away there is another planet with markedly slower orbital velocity around its sun and the planet has a much slower rotation than earth.
If we had two atomic clocks, synched them up here on earth and then took one clock to PlanetX wouldn’t the two clocks eventually start to disagree on what time it is? I know neither time is preferred or more correct than the other one but nevertheless I thought movement did determine the rate at which clocks moved relative to another clock moving at a different rate.
(FTR I do realize that both earthlings and PlanetXians would perceive their time to be perfectly normal even if the other planet viewed them as moving somewhat faster or slower in time.)
Well, The answer is no if you read too quickly and miss the words I bolded below.
Jeff_42 is correct. The answer is yes.
Thanks for the responses so far, guys.
Perhaps the question was badly worded. If we look at the universe as a single object, is it possible to be completely still, relative to that object? Can you say “relative to the universe, I am not moving at all”? And since everything within the universe is moving, would you be able to tell?
As Timothy Campbell and DrMatrix said, there is no absolute reference frame you can use. There is no “outside” to look in from and consider the universe as one object. From here inside the universe, you gotta deal with Relativity. It’s a profound (and as you’re finding, counter-intuitive) conclusion of Einstein’s theories.
Physics is weird that way. So are some physicists 
And there’s the crux of the matter. There is a thought experiment about that. Imagine that you are the only object in the universe, you alone in a totally black void (in a spacesuit presumably, because this is a total vacuum). Can you judge if you are in motion? No, because you have no other objects in the universe to measure your motion against. Can you even measure your mass? No, not without another object to compare against. You don’t have to measure your mass by gravity, you could use an inertial measurement against a counterweight (that’s now astronauts do it). But there is nothing else in the universe to measure against, just you. You have no measurable mass. If you ever find yourself in these conditions, please do us all a favor and go Big Bang. The universe and all of space-time depends on it.
Well, there you go. You’re imagining a universal frame of reference of “the whole universe” to measure the total motion of the universe relative to you. This is the only way anyone could concieve of such a thing, and it screws the physics so badly that you can’t reduce it to an answer. It’s a fairly complex thought experiment, I’m sure I mangled it a bit, but if you want to read it yourself, it’s in “Relativity for the Million” by Martin Gardner.