Basically you are right, but in this context (in this word order), it sounds a little unnatural or "overemphasized", if you will; which is related to the article.
In addition, "als Kind" means/includes the entire childhood > "als Kind" = 1. "als ich ein Kind war." / 2. "in einer/meiner Kindheit" > "as a child" = 1. "when I was a child" / 2. "in someone's / my childhood".
Which is thus only a shorter substantivized form.
It should be: "Die Kinder sind lauter als ich."
The explanation -- I can think of -- is, "als" is in this case a conjunction which precedes a shortened relative clause: "Die Kinder sind lauter als ich es bin". So the pronoun in question has the function of a subject, that's why it should be in nominative.
(Please correct me, if i made any mistakes)
Does German have distinct adjectives for loud and noisy? Or is it all 'laut'? Loud describes the sound as having high amplitude. Noisy is not precisely defined, but typically refers to sounds extended in duration. For example, an air horn deployed near your ear would be loud, whereas a dog that's been barking for the last two hours is noisy.
die is used before feminine nouns and also before plural nouns.
Here, it's used because Kinder is plural.
(There is no gender distinction in the plural in German -- all nouns take die in the plural regardless of whether they are masculine, feminine, or neuter in the singular.)