What device/instrument of war and/or military practice in our military today is a derivative of ancient Rome?

Answer 1

No weapons, other than maybe artillery from the Roman Ballistae and Onagers, but the structure of the armed forces, and some of their practices, can still be found. Namely:
1- Organization into "squads" of 8men who, messed, slept, trained and fought together. Centuries/Cohorts and Legions were built upon this basic structure. What the Romans discovered was building cohesion like this, make for a more effective combat unit
2- Professional Non-commissioned Officers. The Centurion was roughly the equivalent of a Warrant Officer or senior NCO. They were selected solely for their leadership and combat experience.
3- Specialized units. The Romans had dedicated engineers, medical, and administrative personnel attached to each legion.
4- Individual combat loads/marches. Under Marius' the Romans lightened their baggage train by making each legionary carry his own equipment. What they discovered was the maximum a soldier could carry was about 60-70lbs. This was later written into military regulations. The US Army expended several million dollars in the mid-80s to find out what the ideal load of a soldier should be: Yep it turned out to be 60-70lbs. The Romans also saw that the maximum daily march to expect from one's troops was about 20miles. That was the standard until the advent of mechanization in the mid-20th Century.
5- Professional troops and pensions. Yes,they also found that professional fought better than draftees. The flip-side is one must provide incentives to stay in. For the Roman Army this was a pension/land after 25 years service, and citizenship for non-citizens serving.

Answer 2

Roman Instrument

Answer 3

Some of the principles of navigation and trajectory are derived from ancient times.

Answer 4

Almost all of it.