I have just received my J-1 visa (short-term scholar). On the visa it says "Expiry date: 14 November". My flight back is on the 20th of November. Do I understand correctly that even though the visa expires on the 14th, I get additional 30 days to leave the country? Thanks in advance for your answers.
A US visa is solely for entry. The visa's expiration date is simply the last day you can use the visa to enter the US; it has nothing to do with how long you can stay in the US. So for example, you can enter on the day the visa expires, and stay for potentially years, if you remain in a valid J-1 program that whole time (with a valid DS-2019 and remain in the designated program).
J-1 status lasts for 30 days after you complete the program. So it depends on when your program completes, which you have not stated.
Upon completion of your exchange program, you have a grace period of 30 days to depart the United States.
Note that typically you are not allowed to re-enter in those 30 days - even across the border from Canada or Mexico.
If your J1 is expiring you don't to worry much you can still work in US on sponsored E2 non investor visa. I know ALTEN Calsoft Labs offer E2 visa for French nationals