The date your College declares you to have passed the course , you need to stop working .
Based on your facts you have completed your studies on 3rd Jan .... unless your college gives a letter stating that you have completed the program which is dated before that date .
1. Coop done before the date of completion should not be a problem. You can write the actual dates the university authorised you to do the Coop. It is better to write University as the employer in the main application and in the explanation give the actual employer names and dates.
2. Till you pass, you are allowed part time ; Which you have stated to have done. But 3rd Jan is the grey area , which may create problem for you. You cannot work after your transcript comes out ( or passing of course whichever comes earlier ). However this employment you can write on the main application and give the nature of job in the explanation .
You can again start working moment you apply for PGWP .
This is my personal opinion and it would be better to consult other seniors who are definitely more knowledgeable on these matters. Best wishes.
1. I have got my graduation letter stating my course completion is Dec 13,2019 ,but coop ended on Dec 20 ,2019
2.What start date I should mention in the job offer section in pgwp application
3.Should I mention the current employer - the company I started to work from dec 23 in employment section
You cannot be on coop when your course is completed . That portion would not be a coop job . However bulk of the University that I know never end a program on Coop . End term is always academic .
In your case you need to talk to your college as to the difference between last day of the term ( assuming it to be 13th dec ) and program completion ( transcript date ) . After all you can fail your coop also ( again assuming that there is a grading given by coop employers ) .
With the current completion date , you may be in trouble . You will need to write a full explanation as to how you took the program completion as the day of transcript since you could technically fail.