Does the Milky Way's movement through space have spacetime implications? The way I usually see spacetime explained, it says that if you just sit still, you are moving at 0kph and so you are moving at maximum speed through time (ie. time for you is flowing at maximum speed).  However, sitting still we're moving at the speed of our galaxy through space, which is about 600km/s.  The speed of light is about 300,000km/s.  So for simplicity's sake, let's say the galaxy is actually moving through space at 1,000km/s.  Does this in fact mean that if I sit still on Earth, time for me is actually flowing 1/300th slower than if I were sitting still with respect to the universe?  And does it also mean that if the Milky Way were travelling at 99% the speed of light, time for me would be flowing at 1% the rate of the flow of time for someone that was sitting still with respect to the universe?
Your question has a misunderstanding at its root:  You say, "So for simplicity's sake, let's say the galaxy is actually moving through space at 1,000km/s". The problem here is that the whole point of Relativity is that the idea of movement with respect to space is meaningless -- the only thing that has meaning is movement with respect to other bodies.

Time dilation is also relative. If you're moving at 1000kps relative to Fred, Fred sees your clocks as running slower, and in his "reference frame" they are.  But you, in your reference fame, see Fred's clocks as running slower! (This is the essence of the famous Twin Paradox.)

You're being led astray by your intuitions about space and time. (And don't be ashamed about that. Newton, who was probably the greatest physicist ever, even greater than Einstein, and Kant (one of the great philosophers) and many, many others had the same intuitions. Yet experiment has taught us that they were wrong.)  Neither space nor time is absolute, and an understanding of Special Relativity (which is much easier to understand than it was to discover in the first place!) is required to ask as well as to answer your question.