So if you're saying "No, I dont live in Osaka" would you then be able to say いいえ大阪に住んでじゃない or do you have to say 大阪に住んでいません? Also if you can't just use it whenever, why not? Is じゃない reserved for talking about a verb, noun, etc. or is it just something that turns almost anything into a negative?
じゃない is only for nouns and na-adjectives. (Na-adjectives are technically descriptive nouns, so you can say じゃない is only for nouns and noun-likes).
I'm not a student.
This question isn't easy.
じゃない is an informal short version of ではない, which is a plain version of ではありません. All these three are for nouns and na-adjectives.
You cannot use じゃない to negate a verb or an i-adjective. A plain (non-polite) version of 大阪に住んでいません is 大阪に住んでいない, or more colloquially, 大阪に住んでない (でいる can contract to でる).
(If you are already familiar with explanatory-の, you can use じゃない in conjunction with it to effectively negate a verb: 大阪に住んでいるんじゃない. But this ん changes the nuance of the sentence quite a bit, so make sure you understand what you are doing before using this pattern.)