Yes it does. There have already been about 4 or 5 proofs/demonstrations that show this.
It seems your flaw of thinking is that with 0.9999.... the 9s end at some point. Well they don't. They keep on going for infinity. If they did end at some point, then yes, you could say "it gets close but does not equal 1." But since they never end, it is equal to 1.
Look, no one ever said that it would be easy....
No, it doesn't. It's as close as you can possibly get to 1 without actually getting 1. Also, whoever posted 9/9 = .999... is an idiot. Morons.