1 is done ususally with the aide of CPS due to neglect/abuse. They are required in all states to have laws reguarding children in state care for 18 months or longer.
2 in divorced situations, when 1 parent has abandonded the child, there may be a way to have a parents rights removed.
However, this varies, sometimes Child support still continues, sometimes it takes both lack of financial support and physical presence. Sometimes it requires there be an adoptive step parent.