Can a mother who constantly insults, derogates in front of others, slaps, redicules, and makes her teenage children cry be held as emotionally abusive and thus an unfit parent?

You said "slap" that is physcial abuse.

It is grounds if the judge accepts it, some judges may accept it, others may require parent to take a parenting class. There is no "for sure" ever in family court

Are you trying to get custody of the child?

With this behavior from the mother, my sister, yes. They do not have a father as he is married and living in a different state and has never had anything to do with them since they were born.