1 != 0.99...

Consider the floor function. It ignores the decimal part (everything right of the decimal point) of a number.

Thus, floor(0.999...) = 0, and if 0.999... = 1, floor(1) = 0.
However, floor(1) = 1, therefore we must conclude that either 1 = 0, or the hypothesis was wrong.

>>1
Your mistake is identifying the reals with decimal representations while allowing infinite sequences of 9's. If you allow infinite sequences of 9's, decimal representations become non-unique. In particular, 0.\bar{9} and 1 represent the same real. This also makes the intuitive mechanism of some functions, like the floor function, break down.

>>3
There are no excessively abstract ideas involved in constructing the reals, and after learning any such construction one understands why 0.\bar{9} = 1. Considering all modern engineering is underpinned by calculus, which is rife with ``notational bullshit that doesn't make sense,'' it seems that perhaps it does make sense after all.