3 questions (Copenhagen)

Hi ,
we are a young couple with a baby who is less than a year old. We are non-EU citizens, and considering moving to Copenhagen for work. However we are looking for answers to a few questions:

1. We intend our baby to start school in Copenhagen from the start. Would public schools be free of cost for us?
2. Need details about "Live-in nannies". Do you get one from nanny agencies ? How much they charge on a monthly basis?
3. Is medical free of cost for the whole family ?