Is it permissible to pay off my deceased mother's debt from the zakat money sent to me, and is it permissible to give charity to homes for the elderly and orphans, and is it permissible to give charity to my family members who have a modest income?

You are not permitted to spend the money sent to you unless the donor specified particular uses for it. If it is zakat money, you may give it to the poor and the needy or any of the eight categories. If it is a voluntary charity, you may give it to whomever you deem worthy.