Re the separate question of start dates and notification, please vote for this:
Re tasks assigned to you by others without due dates for which it’s not appropriate for you to add them:
My approach here is to create a private follow-up task Shift+Tab+F for me with the date I want, and just move the original task to Later without a Due date because I know I have it covered with the follow-up. (For my own tasks, I might sometimes create a subtask instead of the follow-up task similarly.