Short answers, pulled from the story.
Max Wolf discovered the trojan asteroid 588 Achilles on the 12th of February 1906. He used photographic plates at Heidelberg Observatory to find this object. This discovery marked the first time Lagrange's theoretical calculations applied in practice within our solar system.
Asteroids orbiting near the L4 point receive names from the Greek side of the Trojan War. Those near the L5 point take names from the Trojan side. Two exceptions exist: 624 Hektor in the L4 group and 617 Patroclus in the L5 group were named before the convention was adopted by astronomers.
Only nine Mars trojans have been found to date. Thirty-one Neptune trojans share that planet's orbit. Two Uranus trojans, two Earth trojans, one Saturn trojan, and a temporary Venus trojan remain documented.
All known trojan moons belong to the Saturn system. Telesto and Calypso act as trojans for Tethys while Helene and Polydeuces serve as trojans for Dione. These small satellites occupy stable positions near Lagrangian points within the planetary system.
The system likely stays long-lived if m1 exceeds 100 times m2 which exceeds 10,000 times m3. Here m1, m2, and m3 represent masses of the star, planet, and trojan respectively. A formal stability condition requires 27(m1m2 + m2m3 + m3m1) to stay below (m1 + m2 + m3) squared.