I'd have to see TheEngineer's estimates of the velocity of the balls. Just judging by the angles involved, energy is indeed conserved.
You can't disregard the fact that, using a well-designed apparatus, the value of G is consistent. Just pointing out that people have gotten wildly different values doesn't mean anything because these outlier values were obtained using faulty apparatus that probably had the very problems that you are describing... Explain to me, how could hundreds of measurements, using non-identical apparatus and parameters, produce the same answer, within a 1% error, stipulating that the apparatus was built by a competent engineer? If you still insist that the values obtained are so different as to preclude the mere possibility of correlation, please point out to me one of the experiments performed which indicate an inconsistent value, so that I can evaluate it for myself? (although I must point out once more that the existence of a correlation much looser than the slim errors found in the value would still constitute valid evidence toward the existence of a gravitational constant. the errors would simply be showing the inconsistency of the apparatus)
Miles Mathis' paper is just ridiculous. I skimmed over the whole thing... first off, not once does he mention the strong or weak force. Just do a search for "strong force"... nothing. He does mention some completely incorrect ideas about E/M fields affecting the attractions, but again, it is obvious that he just doesn't understand the underlying theories of E/M. His idea that EM actually is part of the "pseudo-field" is completely wrong. Seriously... how could you believe even a word of that paper?
I have to inquire, Tom... what is the highest level of mathematics and classical physics that you have actually learned? Based on your acceptance of Mathis' paper, I would estimate that you have only the most basic introduction to classical physics, probably only learned up to Calculus I, and have obviously never heard of a vector. If I'm wrong here, let me know, so I know what level of technicality I can include in my responses.
I'm not going to refute everything in Mathis' article, because literally 95% is just meaningless drivel. Mathis doesn't even have a technical degree... while I admit that degrees in themselves don't mean much, his actual formal education has been limited to Art, Politics, and Management. Hardly a trustworthy source... and have you looked at his other papers? In one of them, he claims that the constant pi is actually not dimensionless, has the units of velocity, and is actually closer to 4 than 3.14. He doesn't even know the difference between velocity and speed.
Since I know you won't be pleased about my utter disregard for anything Mathis has written, I'll respond to the one part of the paper which makes physical sense, but is mathematically incorrect in its interpretation: the idea that the walls of the apparatus would have affected the calculation. It is true that under the accepted theories of gravity, the walls would indeed exert a gravitational force on the balls. However, even if the walls were 1000 times the mass of the balls, negating any inverse-squared-distance argument I could make regarding the attenuation of the fields produced, the walls would STILL not have affected the experiment. If you examine the apparatus described by Cavendish, you see that the line between the centers of mass of the two balls is parallel to the nearest wall. As such, the force does no actual work on the balls, because the balls are constrained to move in a single dimension - or at least, they are constrained to a single degree of freedom which is orthogonal to the wall's field.
So, here's my conclusion: The Cavendish Experiment is valid. The wall's affect is negligible, both by design of the experiment, and because of the significant difference between the distances involved (the distance of the non-parallel walls negates any affect they would have). Mathis' analysis is just wrong... he has the foundation of a high schooler, and his axioms are just plain old false. Furthermore, Mathis obviously doesn't have any type of formal training in these types of analysis... he isn't knowledgeable enough of the theories of classical physics to properly refute them.
The idea that either EM or strong/weak forces could affect the apparatus is also false. Yes, even at the 10^-10N levels that Mathis constantly brays about. Also, strong/weak forces are NOT macro forces... they would NEVER be observed at greater than the molecular level, and Mathis doesn't even address them. Strong/Weak forces are called Nuclear forces because that is where they operate... on the nucleus of atoms.
You keep referencing sources which are just plain old unreliable... While I admit that I don't have all of the data I would need to evaluate TheEngineer's analysis, it appears at the surface to be invalid.
And just for the record, Mathis believes in gravity. He just doesn't accept the validity of experiments which measure the Gravitational Constant.