Can we submit the Duke specific questions after we submit the Common App and still be considered for the Dec 20th priority interview deadline? Or must everything be submitted before the 20th?
Yes, you can submit supplemental items after the 12/20 date and still be considered for the priority interview deadline. The “date received” in the system is based on the common app submission.
Note that some areas still will offer interviews to those who submit after Dec 20. In my area of the country, we have plenty of alumni, so there is often room to get all of the interviews done.
So just it make sure, as long as my common app essay and activities part is submitted i can wait until Jan 1st to submit the rest?
Jan 2nd. 
Look here:
http://admissions.duke.edu/application/checklist
Call admissions if you’re unsure of things – they’re friendly and would have more definitive answers than somebody on this site. But, yes, if you’re in an area with a lot of alumni, you’d probably get an interview regardless of submission date.
Agree with bluedog