Moi non plus !

Because moi is the stressed/disjunctive pronoun, not the normal subject pronoun.

• je = I as the subject of a full verb: Je parle.
• moi = me / I, used for emphasis, contrast, or short standalone expressions

In Moi non plus !, the verb is understood, not spoken. So French uses moi, not je.

Compare:

• Je ne veux pas y aller. = I don’t want to go.
• Moi non plus ! = Me neither / Neither do I.

Yes, exactly.

• Moi aussi ! is used after an affirmative statement.
  
  • J’aime le café.
  • Moi aussi !
• Moi non plus ! is used after a negative statement.
  
  • Je n’aime pas le café.
  • Moi non plus !

So they are a matching pair:

• aussi = also / too
• non plus = not either / neither

Non plus is the part that gives the idea of neither / not either.

It usually appears in negative contexts and means something like also not.

For example:

• Je ne veux pas y aller non plus. = I don’t want to go either.
• Moi non plus ! = Me neither.

So in this expression:

• moi = me / I
• non plus = neither / not either

Because this is an elliptical expression: French leaves out the part that is already understood from the previous sentence.

Example:

• Je ne suis pas prêt.
• Moi non plus !

The full idea would be something like:

• Moi, je ne suis pas prêt non plus.

But in real conversation, French usually shortens it to Moi non plus !

No. Je non plus is not correct.

If you want the short reply, use:

• Moi non plus !

If you want a full sentence, use a normal negative sentence with a verb, for example:

• Je ne veux pas non plus.
• Je ne l’aime pas non plus.

So:

• Moi non plus ! = correct
• Je non plus = incorrect

Use Moi non plus ! as a short reply when the rest is obvious from context.

Example:

• Je ne comprends pas.
• Moi non plus !

Use a full sentence when you want to say the whole idea clearly or add information.

Example:

• Je ne comprends pas non plus.
• Je ne veux pas y aller non plus.

So:

• Moi non plus ! = short conversational answer
• Je ne... pas non plus = full sentence

It is pronounced approximately:

mwa non ploo

More carefully in IPA:

/mwa nɔ̃ ply/

A few notes:

• moi sounds like mwa
• non has a nasal vowel; the n is not fully pronounced like English non
• plus here is pronounced ploo; the final s is silent

So not ploos here, but ploo.

It is neutral and very common. You can use it in everyday speech and in normal writing such as dialogue.

It is not especially slangy, and it is not overly formal either.

You might hear it:

• in casual conversation
• in interviews
• in films and books
• in ordinary polite speech

If needed, you can make it sound a little softer or more polite with intonation, but the expression itself is standard.

Yes, as long as you mean that the same negative idea is true for you too.

Examples:

• Je ne sais pas.
  Moi non plus !
• Je n’ai pas faim.
  Moi non plus !
• Je ne suis jamais allé à Paris.
  Moi non plus !

It works with different kinds of negation, not just ne... pas.

The key idea is: someone says something negative, and you agree that it is also true for you.

Yes. French often uses intonation, pauses, or extra words for emphasis.

For example:

• Moi non plus !
• Moi non plus, franchement.
• Eh bien, moi non plus !

You can also use moi to create contrast:

• Lui, il aime ça. Moi, non plus !
  This is less common in that exact setup, but it shows how moi can be emphasized.

Very often, though, simple Moi non plus ! is enough.

No. It depends on tone.

• Moi non plus. = neutral
• Moi non plus ! = more lively, emotional, or emphatic

In speech, the difference is mostly in intonation. In writing, the exclamation mark just shows stronger feeling or energy.