Is this an idiomatic sentence that's commonly used? It's in the "Phrases" lesson but it doesn't sound like a common phrase to me.
It's perfectly fine.
Fritz: Kommt mal her!
Hans: Ich auch?
Fritz: Nein, du nicht.
Eh yo, "Nicht Sie" is also accepted as a legit answer; question is, is it grammatically correct?
This one was 'nur ein bisschen' strange for me. Let me ask you a question, if I wanted "Du nicht" in the end of a question, like we do in english "bla bla bla, aren't you?". Would I say "nicht du?" or other word. Once I saw the word "oder", this would work as the "aren't you?" "isn't it?" and so on.
You can't use it as a question tag. You could use "oder?", "nicht wahr?" or "nicht?".
"Dich" is for a verb. It's used tu say a verb applies to "you". For example, "Ich liebe dich", I love you. The verb "lieben" is made by "ich" to "du". Then you say "dich".
Other example: "Ich esse dich." (I eat you) "Ich" makes the action, "du" receives it.