Can a mother use a different surname for her child at school, or to register with the doctor, hospital, etc. without the consent of the father (who has parental responsibility) ?

Not at hospitals and school registrations, or doctors. You can call him whatever you want and tell the school that you would prefer him to be called whatever you want but the legal documents have to be in his legal name.

The idea behind using another name is that there is no intent to defraud. In the areas your trying to make the changes then it would be an attempt to defaud the ex by way of a name change and would be considered illegal.