Can a beneficiary with a criminal history apply for a green card?
The U.S. government typically asks to know about every interaction an applicant has ever had with law enforcement, both in the applicant’s home country and in the United States.
In general, the only exception is traffic violations.
For more information on the impact of a criminal record on a green card application, see this article.
If I am out of status, can I use the travel permit to leave the United States while my marriage green card application is pending?
U.S. Citizenship and Immigration Services (USCIS) doesn’t explicitly forbid green card applicants from traveling while waiting for their application to be processed. However, it is up to the discretion of the Customs and Border Patrol (CBP) officer to deny the individual’s re-entry into the United States. If possible, it is preferable for an applicant to wait until they receive their green card before leaving the United States.