Must I seek my mother's permission before giving some of what she bought me to my friends, or is it permissible for me to give it to them secretly, knowing that she forbids me from some desirable Sunnahs, such as shaking hands and spreading greetings of peace?

The virtues of generosity and altruism are noble, and Islam encourages them. However, you are not permitted to donate tools that your mother bought for you, especially if she forbade you from doing so. You must obey her in matters that do not involve disobedience to God. Your altruism towards others is not obligatory, and it might harm your mother if you replace what you donated. As for shaking hands and spreading greetings of peace, if no harm results from them, then it is preferable to spread them and not obey your mother in that regard. We advise you to obey your mother in matters that do not involve disobedience to God.