You should not be able to assign a task to an Inactive member because they are inactive (they should not be showing up in the assignee options window). If you are able to assign a task to an inactive member, how are you doing this? From the Bulk assign, by editing a task, through Importing tasks, or Project Syncing using ProjectTransit? I will repro and see what is happening if you could provide me with this info.
Thanks!
p.s. for active members there is no way to shut of the assignment notification as a manager. Only users have the option to disable the email portion.
Steve Thompson | Solutions Consultant
Daptiv