Will the children be sinful for not teaching their mother how to read and write and some short surahs, and for not knowing how she prays, and what should they do now? Is it haram for the mother to kill a cat that was harming her chicks, and what should she do to atone for that?

We thank the questioner for his concern for his mother, and we advise him to teach her Al-Fatiha, because prayer is not valid without it. Whoever falls short in this while being able to do so is sinful and must repent. As for his mother killing a harmful cat, there is nothing wrong with it as long as the harm cannot be repelled except by killing.