Learn Italian: Non sono mica sordo, non devi gridare. - I am not deaf at all, you don't have to shout.

Using mica adds strong emphasis, meaning 'at all' or 'it is not like'. While Non sono sordo is just a neutral fact, Non sono mica sordo directly reacts to the other person's behavior, essentially saying: 'It is not like I am deaf, so why are you treating me like I am?'

Can I put mica before the verb instead?

Yes, you can! But if you move mica before the verb, you must drop the non. So, you could say Mica sono sordo, non devi gridare. This is a very common structure in everyday spoken Italian and means the exact same thing.

Does non devi gridare mean 'you must not shout' or 'you do not have to shout'?

It can mean both! In English, 'must not' (a strict rule) and 'do not have to' (a lack of necessity) are very different. In Italian, non dovere handles both concepts. You have to rely on context, and here, 'you do not have to shout' is the most natural fit.

How would this sentence change if a female speaker said it?

The adjective sordo (deaf) must agree with the person it describes. If a woman were speaking, she would change the ending to match her gender, saying Non sono mica sorda.