Is it permissible to perform Umrah on one's own behalf and dedicate its reward to more than one deceased person, and is that considered fulfilling the trust of someone who requested performing Umrah for his deceased mother?

You may perform Umrah on behalf of your friend's mother if she has not performed it before, is physically unable, and grants you permission. However, it is not permissible to include anyone else with her in a single Umrah. As for performing Umrah for yourself and sharing its reward with others, it is permissible without limitation to the number of people you include, whether they are alive or deceased.