1. Yes, this is the default behavior of the Tasks page in PWA, and your team members only need a Project Essentials license to do this. This is assuming that the Project Online administrator has not "locked" the Start and Finish columns for editing. The locking of these columns is done in the settings for the My Assignments view, by the way.
2. Yes, this is entirely possible. However, the new settings will only apply to unstarted tasks. Tasks that are in progress will continue to use the old method.