Learn Italian: Non sono mica pazzo, dico la verità. - I am not crazy at all, I am telling the truth.

It makes the negative much stronger. While non sono pazzo just means "I am not crazy", adding mica translates closer to "I am not crazy at all" or "It's not like I'm crazy!" It is often used to defensively contradict something the listener might be thinking.

Where exactly does mica go in a sentence like this?

It usually goes right after the verb. The standard pattern for this type of emphatic negation is non + verb (sono) + mica.

Could I drop the non and just use mica?

Yes, but you would have to move it to the front of the sentence. You can say Mica sono pazzo for the exact same meaning. If the word stays after the verb, the non is absolutely required.

If a woman is speaking, does anything in the first half change?

Yes. Because pazzo (crazy) is an adjective describing the speaker, a female speaker would change the ending to agree with her gender, saying Non sono mica pazza.

English uses "I am telling", but the Italian uses the simple present dico. Why not sto dicendo?