Yes, read the brochure/booklet given at the embassy on the date of your interview!
A J-1 student has 1 month to travel throughout the country after the end date on their permit. You can stay in the states for this period; however you are not eligible to be working!
Yes and No, it depends upon the contractual agreement between your employer and your sponsor pertaining to the job!
No! You are not eligible to work on your tourist visa. Also you are only eligible to work with your social security card with the issuance of a valid DS2019 permit/form was issued for.
Well, unless you have a very valid reason to excuse your lateness. It’s a high possibility that you will be denied on your next J-1 visa application. (NB: If you are still eligible for a J-1 visa).
Yes, to some extent. However it’s at the consoler’s discretion!