Is the mother sinful if she stops helping her needy neighbor, after the questioner's grandmother had advised her to give charity and help the needy? And is the questioner obligated to help this neighbor after her mother's death, whether the mother had advised that or not?

Helping a neighbor is a good deed that is rewarded, but it is not obligatory. If someone more deserving of help is found, then it is better to direct assistance to them. There is no sin in ceasing to help her without a reason, but it does mean missing out on the virtue of charity. It is advisable to continue helping if it is highly probable that she is deserving. Upon the mother's passing, there is no obligation to continue helping her, and there is no sin in ce ceasing to do so; because it is a voluntary act for which one is rewarded, and there is no obligation to continue it.