Answer: Yes. Consult an Immigration Lawyer for advice specific to your situation. There is what is known as a K-Nonimmigrant as the Spouse or Child of a U.S. Citizen.
The Legal Immigration Family Equity Act
And its amendments (LIFE Act),
Established a new nonimmigrant category within the immigration law,
That allows the spouse or child of a U.S. citizen,
To be admitted to the United States in a nonimmigrant category.
The admission allows the spouse or child to complete processing,
Permanent residence in the United States.
To have permission for employment
to permanent resident status. An Immigration Lawyer can help navigate these categories efficiently.

Answer:
A person may receive a K-3 visa if that person:
Has concluded a valid marriage with a citizen of the United States;
Has a relative petition (Form I-130) filed by the U.S. citizen spouse for the person;
Seeks to enter the United States to await the approval of the petition, And subsequent lawful permanent resident status,
and, has an approved Form I-129F, Petition for Alien Fiancé,
Forwarded To the American consulate abroad where the alien wishes to apply for the K-3/K-4 visa.
The designated consulate is the one with jurisdiction over the current residence of the alien spouse. Consulting an Immigration Lawyer can assist in determining eligibility.

Answer: Persons in K-3 or K-4 status
And applicants for adjustment to permanent resident status,
From K-3 or K-4 are eligible to apply for a work permit,
An Immigration Lawyer can help clarify eligibility for a work permit.

Answer:
You may travel using your unexpired K-3/K-4 nonimmigrant visa
to travel outside of the United States and return.