Learn Turkish: Yine mi? Hadi git!

The question particle mi follows the exact word you are questioning. Here, you are just asking "Again?", so it follows yine. Since yine ends with a front vowel ('e'), vowel harmony makes the particle mi.

What is the dictionary form of git?

The infinitive form is gitmek (to go). To give a direct, informal command to one person, you just drop the -mek or -mak ending. This leaves the bare verb root git, which acts as the imperative command.

Since we are learning negative imperatives, how would I change this to say "Don't go!"?

You would add the negative suffix -me directly to the verb root, making it gitme!. We use -me instead of -ma because the vowel 'i' in git is a front vowel, so it needs a matching front vowel suffix.

How exactly does hadi work in this sentence?

Hadi translates to "come on" or "let's". It is an exclamation commonly used in everyday spoken Turkish right before a command to show urgency, encouragement, or impatience.