I chose for the second question “A new instance appears and the new instance has the date set to the next date (from the completion date)” but I’d like the option of both from the completion date OR from the due date.
The best UI I’ve seen for creating recurring tasks is in the Tasks app for Android (previously known as Astrid Tasks).
It packs all of the essential options into just three lines (due date/time, repeat by X Y, repeat type):
Video demo (really helpful to see the extent of “repeat by X Y”)