1. (a) That's wrong. Battery consists of any intentional or negligent act that results in some physical contact with a plaintiff's body. But it is also a requirement that the contact take place either with a hostile intent, or without the plaintiff's consent. Battery is an instance of trespass to the person.
There are many circumstances in which one person freely allows another to make physical contact with their person, for example, a visit to the doctor or hairdresser. Another obvious example is participation in contact sports. In such situations, as long as the contact takes place within the rules of the game, the plaintiff will have consented to such acts and there will be no action for battery.