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If A=B and B=C, then A=C > TRUE.
I can't be bothered to read the whole thing, so if there is a particular statement or idea you want to address, you can always let me know.
I will comment on the top part:

If A Equals B, and B Equals C, then A Equals C
But there is a problem, which is this:
For A to be A, and B to be B, there must be something about A which is different from B. Therefore, A cannot be precisely equal B. Therefore, the equation can never be precisely true.

So? Just because there's something about A that is different from B does NOT mean that A cannot equal B.
For instance, you have 3 piles of coins:
Pile A: 1 quarter, 2 dimes
Pile B: 4 dimes, 1 nickel
Pile C: 3 dimes, 2 nickels, 5 pennies
All piles contain 45 cents
So mathematically they are equal, even if they are different.
Besides, from a quick read, it seems the author is talking about cause and effect (smoking and SIDS). This has nothing to do with equality. And nothing to do with equations. We can't even use mathematical logic (i.e. if A implies B and B implies C then A implies C) because smoking does NOT imply SIDS (otherwise there would be many more dead babies). This is more in the field of probability than algebra.
It's true. Transitive rule.