Quote - he speed of light is critical in three equations, the Lorentz transformations.  They're the ones that show that, as you go faster time seems to slow, mass increases, and length decreases.

That's the Lorenyz theory that later Eistein put as his Relativity theory. This is a theory and if you apply it is impossible to achieve light speed or higher, but first we must understand aether1s Lorentz theory.
Lorentz theory is based on Maxwell electromagnetic theory. Maxwell summed in a brilliant theory the worjs of Ampere and Faraday, the problem of Maxwell's theory is was that it was made only for stationary bodies, bodies that don't move and of course no force on a moving charge was defineded.
Yje consequence is that when we try to apply Maxwell's equation to a moving body we find that the form of the equations is not preserved (covariance). In other words the Maxwell's equations in Earth and Mars are not the same, something not good.
To overcome this problem Lorentz found a tranformation of coordinates that preserve the form of Maxwell's eauations in any system. In these tranformatins time is not more absolute and must transform too.
But this is only Lorentz approach to solve mMaxwell's equation problem. There were other solutions for the same problem that most of people doesn't know that exist.

• Hertz solved the problem replacing the partial time derivates in Maxwell's equations by total time derivates.
• Weber created other different electromagnetic theory that was based on the Ampere's force (not the force found in school books that is Biot-Savart force and not Ampere, Ampere's force is not in school books!!!!).
  If we apply anyone of these theories we have the same practical results and equations, but there is a big conceptual difference.
  On Çprentz theory we cannot achieve speed of light or overcome becaise time becomes infinite or imaginary.
  In Hertz or Weber's theory the force acting on a charged moving body becomes zero and it doesn't respond anymore to our electromagnetic fields. If the body is traveling faster than light the direction of the force is reversed slowing the object. Anyway there is nothing impossible for something move at light speed or above.
  In anyway these three theories can only be applied to charged bodies and never to neutral body.

Quote - These have all been experimentally demonstrated.  In fact, GPS satellites have to take this into account because their clocks tick at a slightly different than does a clock on the surface of the planet:....

He, he, he. .... GPS doesn't use Relativity and if use the results must be corrected by the Sagnac effect
What is the Sagnac effect? First at all what is an effect?
An effect is something that is observed, has empiric equations, but cannot be explained by any theory. Many 'effects" today have theorical explanation, but still are called  "effects''.
To understand all we must begin with the failed Michelson=Morley experiment (the failed one!).
Michelson-Morley tried to measure the absolute speed of Earth using light. They tried and failed.
They were not able to measure the absolute speed of Earth. To explain the fracass Einstein postulated that the speed of light is always the same in any direction and in consequence that is impossible to know the absolute speed of something. Yhis was the fundamental stone of the theory of Relativity.
This is the story that people know, but waht people don't know is that the story has not ended here.
Einstein's theory was not accepted, ignored or people didn't knew that Einstein existed, so Michelson-Morley continued with their experiments and tentatives and twenty years later had sucess measuring the absolite speed of Earth. Not only Michelson-Morley contined, other people did with different experiments and had success too. Among these people was Sagnac that in the 1920 decade has success. Nobody knows the existenece of those succesful experiments, it wre buried because contradicts Relativity that become a Holly dogma.
Sagnac survived by some very particular reasons even was burries for more than 50 years.
Sagnac's experiment was very simple and reliable. If we replace the light sources used by lasers, use solid state lasers we can have a very small, cheap and without moving parts that is able to measure the absolute speed of something, something that technology and industry cannot resist and so, devices based on Sagnac's effect are a common part in planes, satellites and missiles, they contradict Einstein, but they work and create money.

Quote - Incidentally, the reason why your tomato example in the particle accelerator makes no sense is because there isn't enough energy in the particle acccelerator to move a tomato.

You need very little energy to throw a tomato, any kis can do it and the particle accelerator has mega watts of energy, of course hundreds of thousands much more energy that the hands of a kid.
Is very simple, no matter how much energy you put in the particle accelerator the tomato doesn't respond to that energy, but it respond pretty well to the enrgy of your hand.
On the other side, try to move a beam of electrons with your hand....

Stupidity also evolves!