Are there periods where a bipolar sufferer is 'normal' ie neither depressed nor manic?

Yes: The goal for treatment of bipolar is to achieve as many "normal" days as possible.

Yes: Yes, a good percentage of patients with bipolar i achieve that euthymia ( not manic and not depressed). It requires working closely with your psychiatrist to fine tune and adjust medication in order to reach that goal. Psychotherapy is helpful in almost all cases in maximizing benefit.

Bipolar: Indeed, yes, when responding to treatment.