Does the Army have any fitness teams?

The Army does. It's called the Army World Class Athlete Program.

"The US Army’s World Class Athlete Program (WCAP) allows top-ranked Soldier-athletes to perform at the international level while also serving their nation in the military. Our members train and compete throughout the year – and aim for the Olympic and Paralympic Games."
https://www.armywcap.com/

There are also teams at posts sometimes but these are local unofficial teams that are extracurricular. You have have to maintain your primary duties. If you are accepted into the AWCAP then that is your job and you will PCS to their location in Colorado.

Yes, those do exist. Most bases and large units will have sports teams of various types, sizes etc. Also big Army has an Official boxing team and things like that as well. Most branches have stuff like that.

The Army does have a boxing team, and a power lifting team not sure about MMA. Just Google it and there is info. Most were cancelled this year due to COVID but the teams do exist.