stingc

The point of E=mc^2 is that energy and mass are equivalent. Originally, the point of it was to say that a body's inertia (mass) depended on its energy content.

Now, saying energy and mass are equivalent in words doesn't quite work if you sit down to calculate anything. The two quantities have different units. Energy is expressed in joules, foot-pounds, ergs, etc. Mass is in kilograms or slugs. Equating these things directly is meaningless. If mass and energy are indeed equivalent, you need a conversion factor. This should be a constant of nature, and it needs to have units of velocity squared (you can easily check this by using the fact that 1 joule is by definition 1 kilogram-meter^2/second^2). c^2 is the natural choice.

This is the simplest way of looking at it, but there are others. Einstein certainly didn't arrive at the equation like this. He had to say why energy and mass should be equivalent in the first place. If you want to know why that is, I could refer you to some better (but harder) books than the ones you're reading.

The statement about spheres in clay is to help you understand why energy has the units it does. If you simply accept this, the example is irrelevant. In reality, energy was given its present definition (not that the basics are really new, or were even new 100 years ago) due to many complicated and interwoven discoveries. The concept of energy also encompasses so much more than just motion that I think the example is basically useless. I hope that your books contain a much more substantial explanation of what energy is than that.

Kaku's statement that there is no absolute length in relativity is misleading. He meant that one's most naive definitions of length are not absolute in relativity. Hence the length contraction. There does exist an absolute notion of length in relativity. It doesn't change size depending on how fast something goes (unless the object is physically stretching or compressing in the usual sense).

This is actually quite simple to define. If lengths are always different to people moving at different speeds, just define the absolute length to the be the one measured by someone moving at the same speed as the object itself. You might think that this is cheating, but you can set things up such that anyone will measure exactly the same length without actually changing speed to do so.

As for the why things don't expand, this can be shown intuitively through a couple of diagrams, but that would require too much background for me to describe here. If you're interested, I'd highly recommend that you get a relativity book which uses spacetime diagrams to explain things. The viewpoints given in most popular books - while correct - were basically abandoned by most physicists after 1908 or so. The geometric formulation of relativity is far easier to understand, and makes all the length contraction business really simple. It also removes most of the 'wackiness' and apparent paradoxes of 'traditional' relativity.

I hope that helps. I can answer your general relativity questions also, if you're still interested.