How fast is the Earth moving through space? Not toward or away from Betelguese as in red and blue shifts of that particular star but just how fast are we moving through space in general. Can we look at one part of the sky and see everything red shifted and another part of the sky and see blue shifted and extrapolate the total speed from that (obviously we would need a series of measurements)? Do we know how fast the galaxy is moving, or even the speed that the sun moves around the center of the galaxy? For instance if I'm driving a car east at 60 mph, can we take all those factors, add them together and determine the total speed of me and my car.
One of the fundamental principles of relativity is the fact the speed is relative, which makes it meaningless to speak of something have a speed without also saying what that speed is relative to. The only thing that matters here is our speed relative to Betelgeuse.
Does that combined speed cause a time dialation effect (even a tiny one) on Earth? I know time and mass becomes distorted as you approach the speed of light, but I've never heard how steep that gradient is or if there is a lower limit. Would a hypothetical stationary cup of water cooled to absolute zero experience time differently then a similar cup boiling at 100 degrees (obviously the difference would be very tiny, but would it be there or is there a cut off)?
As a rule of thumb, relativistic effects (time dilation, etc.) can pretty safely be ignored at less than 10% of the speed of light. Here's a graph illustrating how time dilation increases with speed, if you're interested: http://www.thebigview.com/spacetime/tdgraphformula.gif
If the universe is expanding in the sense that there is more space between all particles (this was how it was explained to me: that with each passing moment the distance between all particles increases as the fabric of space-time slowly expands) wouldn't the speed of light be slowly increasing (or decreasing) as well. Would a lightyear 600 years ago be the same as it is now?
The gravitational forces within a galaxy are more than strong enough to counteract the effects of the expansion of the universe, so the distance between Earth and Betelgeuse is completely unaffected by it.