Dan's shoulder patch is for the 1st Cavalry Division, based at Fort Hood, TX. He would be attached to a local unit based in New York.
Dan wears his uniform hat ("headgear") while indoors. Additionally, the Major and Staff Sergeant that bring him his orders are wearing their covers indoors against regulations. Military personnel remove their headgear indoors unless they are armed (such as Military Police). Similarly, Major Savage salutes Dan when they first meet, asking where his salute is. As the superior officer, she should have waited for him to salute first.