Learn Italian: Dovresti restare calmo quando guidi la macchina. - You should stay calm when you drive the car.

Dovere does mean "must" or "to have to" in the regular present tense. But when you put it in the present conditional tense, like dovresti, it softens the meaning to "you should" or "you ought to." It is the perfect way to give advice rather than a strict command.

Why is restare in its infinitive (dictionary) form here?

Because it follows dovresti, which is a modal verb. Just like in English where we say "should stay" and not "should stays", Italian uses the conjugated helper verb (dovresti) followed immediately by the infinitive of the action verb (restare).

If I am saying this to a female friend, does calmo change?

Yes, it does! Calmo is an adjective that describes the person you are talking to. If you are giving this advice to a woman, you need to change the ending to match her gender, so you would say Dovresti restare calma.

Can I just say quando guidi without adding la macchina at the end?

Absolutely. Just like in English, you can simply say Dovresti restare calmo quando guidi (You should stay calm when you drive). Adding la macchina just specifies what is being driven, but it is not strictly necessary.