Why is Cantor's diagonal argument usually gets applied to the interval $[0,1)$ and not $[0,1]$? I noted that many formulations of the Cantor's diagonal argument, using base 10, consider only the interval $[0,1)$.
It's mostly convenience, since you don't want to be worried about the integral part of the real number, and in $[0,1)$ it is always $0$.

Generally the diagonal argument when applied to the decimal base, uses digits other than $9$ and $0$ for the "digit change" anyway, so you're not in the situation where you usually run into $0.999\ldots$ as your diagonal number.

Moreover, note that if $[0,1)$ is uncountable, then certainly $[0,1]$ is uncountable. So that really doesn't make much of a difference. (And besides the point, there is a bijection between the two sets anyway.)
Because they never read Cantor's actual diagonalization proof. It was his second proof of the proposition "there are uncountable sets," and the reason he gave for publishing it was that it was both simpler than his first proof, which used the real numbers as the example set, and that it didn't use that set as the example.

Let me repeat that: Geoff Cantor DID NOT apply diagonalization to the real numbers. He used infinite-length strings of the two characters 'm' and 'w'. For example, "mwmwmw...".

Now, if you replace 'm' and 'w' with '0' and '1', you can think of his strings as the binary representations of every real number in [0,1]. And you are right that it should properly include 1=0.111111... .

The reason that it is usually taught using the real numbers, is because this elegantly-simple proof is "dumbed-down" to be taught to High Schoolers, and they already understand infinite decimal representations. Simplifications include (1) using decimals fractions instead of strings, (2) sometimes ignoring that you need to avoid decimals that end in all 9s, (3) treating the diagonalization process as a sequence of steps instead of as a complete, infinite string, and (4) calling it a Proof by Contradiction when Cantor didn't, and it in fact fails as one (the proof is still valid, but the alleged contradiction does not follow from all of what is assumed, just part of it). 

Unfortunately, most of these simplifications have the negative effect of making inquisitive students doubt it. One issue is the one you raised, that 1=0.99999... . This can be gotten around if you never use '9' as the replacement character, but that is unnecessary if you use Cantor's strings.