The legal protection is for your "person." When does a group of cells become a person? I don't know and neither does the "law." In fact, in some countries and societies you aren't a person until you reach a certain age or pass some rite of passage. In our society, birth guarantees personage. Before that it is very murky.
re: why child support etc.
Because without forcing the father to pay child support, the child and unwed or divorced mother becomes a ward of the state and the state most certainly does not want that. That is the historical answer and still holds a lot of weight in modern times. Actually, IIRC child support is actually a modern creation. In the old days, the father would actually take care of the children if the parents divorced. The wife would be kicked out on the street to go her own way. This is in Britain and the US of course, other societies have chosen to deal with breakups differently.
Child support is probably a modern invention mostly because of the increase in children born out of wedlock. If a child was born in wedlock the father would take care of the child. If the father was dead, the children would either starve to death or the mother and children would live with the grandparents or other relative. It really wasn't a large problem back then. They didn't have welfare, the children wouldn't become a ward of the state. There was usually some extended family to support the children or else they were in so remote an area that they would simply die off or so far removed that no one would care what happened to them.
Then with the invention of cities came the rise in orphans and the problem of orphanages.