Is it permissible for my mother to take my grandfather's silver ring, which he used to wear, after his death, and give it to me, knowing that my grandfather has other daughters and sons?

The ring left by your grandfather is considered part of his estate, and it is owned by all heirs in proportion to each one's share of the inheritance. If your mother took it with the consent of the heirs and they relinquished their claim to her, then it is permissible for her to own it and gift it to you. However, if she took it without their consent, she must return it to the estate, and she is not allowed to dispose of it without the approval of the other heirs.