why does it not make sense?

you have to remember that the 9... indicates there are an 'infinite' amount of 9's, in otherwords, multiplying by 10, does not make it have 'one less' 9 at the end that you would normally assume when multiplying a number by 10, the number of 9's after it, is still infinate, so when you subtract, they completely cancel out, leaving you with an integer value.

normally you would write the above proof as:

x = 0.999...
10x = 9.999...
9x = 9.999... - 0.999... = 9
x = 9/9 = 1

0.999... = 1

if you find it hard to think about that, think about it this way, if they were not the same number, then there would be a number that goes inbetween them, like 1 != 2, because there is for example, the number 1.5 that fits between them, but that isn't the case with 0.999... and 1, there cannot exist a number that goes between them, so they must be the same

In this same way, I suppose that 0.0000000000...001=0 if there are an infinite number of zeros before the 1, although anything with an infinite number is mostly theoretical (with the possible exception of pi).

This thread's been around for a while now, so you can't really complain about its creation.
I may be contradicting things I've said earlier, but I don't mind this thread. The posts are thoughtful, and we haven't descended into flaming, like in the one you linked to.

I think Identity's inequality is correct. It's obvious that no real number x exists such that 0.999... < x < 1, so if 0.999... < 1 then that means that the reals aren't dense. But they are.
Therefore, 0.999... = 1.

0.0...01 would not have a 1 at the end (even though I just wrote it that way!), because we are saying "there is no end to the 0s".

Sometimes during our discussions we have been confusing "infinite repetition" with infinity. 0.333... has an infinite repetition, but it is simply 1/3 which is not "infinite"

Every rational number has two decimal expansions, induced by the metric on the real numbers.

Every terminating decimal has two decimal expansions.  But how is this induced by the metric on the real numbers (absolute value)?

That's just assuming that 1 - 0.999... = 0.  We want to show this.