When Alex comes to open the house door to Sanjay, she has headphones on the neck. Suddenly they disappear, and in the next shot they appear again.
Immigration and Customs Enforcement agents are always seated in a glass booth, not an open desk, as they process people returning to the USA.
Since Gloria is married to an American and planning to take the citizenship test, she is already a permanent resident (green card holder). Permanent residents re-enter the United States in the same line as citizens, so Jay's reason for wanting her to become a citizen so they don't have to go through separate lines makes no sense.