Is it permissible to give charity or perform an ongoing charity (sadaqah jariyah) on behalf of an unknown biological mother, and on behalf of an adoptive mother after her death, and is it obligatory to visit the sister of the adoptive mother as a means of upholding ties of kinship (silat ar-rahim)?

You are permitted to give charity on behalf of your mother, even if you do not know whether she is alive or deceased, and on behalf of the woman who raised you. However, you are not obligated to visit the sister of the woman who raised you, as she is not among your close kin (arham). Nevertheless, visiting her and maintaining ties with her falls under the general principle of maintaining ties with Muslims, and it is particularly emphasized if the intention is to express gratitude.