In ancient Rome the men who fought were either the military or the gladiators.

No, no, and no. Gladiators had nothing to do with the military unless they happened to be prisoners of war. Military heroes were honored and held in high esteem in ancient Rome. The gladiators were at the bottom of society.

Alexander the Great had nothing to do with ancient Rome. He was a Macedonian king and military leader. His conquests were east, while Rome was in the west.

No way

Yes, they had a very advanced one for their time.