"All of our experiments on Earth are done in a reference frame moving with respect to the sun, the centre of the galaxy and so on. Yet the Lorentz transformations make the correct predictions for time dilation as observed on Earth, without taking into account the Earth's motion with respect to the sun."
hum, I see no reason why they should?
but again, what experiment does exactly give the results that you talk about with A at rest and B moving
as far as I get it, when A is at rest and B is moving near speed of light and both have clocks they both will see each other's clock run slower, because of the relativity principle you can ask "who's really moving? Both can say that they are moving near the speed of light in respect to each other", hence for relativity it is vital that both see this slowing down
I don't see how those muons a little further explain this ...
"Thus, for example, we see the lifetimes of muons hitting the upper atmosphere as being longer than when the muons are at rest, but precisely the amount that they would be longer if we did the experiment in a reference frame at rest with respect to the sun."
Don't you mean mesons here? (sorry, I'm not into physics like you!)
"This means that Einstein was correct when he postulated that there is no absolute rest frame. Experiments such as the Michelson-Morley experiment confirm that, too."
I find myself not in the position to talk about whether I find an absolute frame of reference possible or not, want to read more about that first in order to understand this issue better.
Marmet and Haspden and others, on the other hand seem not to agree on the interpretation of the results:
http://www.energyscience.co.uk/tu/tu18.htm
http://www.newtonphysics.on.ca/faq/Michelson-Morley.html
"Marmet is going back to the days when we had an "ether", which was supposed to provide an absolute rest frame."
Fact is that vacuum is not empty and is filled with activity of virtual particles. Whether it has those properties that Harold claims it has or not, I don't know.