Is it possible for owners of tasks to be notified automatically via email that they have tasks DUE in 1 weeks (or any time frame?).  I don't see any provision for this.  I'm thinking an alternative for the manager of a project is to create a view in the Task tab based on Planned Finish date criteria (e.g. Next Week).  Then, manually send out email notification reminders.

You'll want option 1.

1. Project Managers have the authority to set automated notifications that go out based on a set list of criteria.  As a PM, go to your project>Admin>Workflow Notifications>[expand]Tasks.  Set the rule that best applies.  In this case notify assignee so many days before task start date.  These settings can be retained in a template but can not be set by Project Type.
   
2. Throwing this in there for anyone interested:  as a user, you can monitor important tasks however your criteria for automated notifications is much more limited (usually when created, edited, updated, deleted).  Select the Actions button to the right of a task and set notifications.