If a child doesn't follow specific instructions and consequently chooses not to come home for several nights, would a parent be considered as abandoning that child if the parent didn't do everything in their power to locate that child?

A parent is responsible for a minor child. If the child does not let the parent know their whereabouts and the parent does not report the child as missing or a runaway, then they could be held accountable. It may not be considered abandonment, but it could result in legal troubles for the parent.

No, the child would be considered a run away perhaps But this is for from abandoment. Poor parenting perhaps.

Next what is your relationship, Do you want the child arrested ? Do you want a change in custody ?