The information should be updated. Usually the employee updates the employer when he/she receives his/her green card. Employers typically reverify...

You were out of status based upon the information provided, but did not necessarily accrue unlawful presence unless you worked without...

An individual with an I-94 stating L2S is authorized to work without the need for an employment authorization document. You can read more at...

It may be possible. It will depend upon how much longer you have on your J-1 nonimmigrant status and the nature of the job opportunity.

To re-enter the U.S., you will need a valid visa. There is no work around.

You must be truthful. If asked about previous applications, be truthful about why the application was refused.

Yes, but if you move, you must update USCIS regarding your asylum application.

I advise clients to never return to the country from which they sought asylum.

You will need to wait for the Form I-129 to be approved and apply for a new visa.

It may be possible. It depends upon more information about your situation.