To unlock the corresponding Gladiator badge for the Arena, you should complete the indicated task:
Arachnobot: get with the Agent of Discord badge
Button Man Gunner: reach level 20
Cobra: get the Snake Charmer badge in Mercy Island
Fire Thorn Caster: reach level 30
Juicer Sniper: get the Watcher On The Knoll badge in Nerva Archepelago
Longbow Nullifier: win one match in the Arena
Longbow Rifleman: get the Locked And Loaded badge in Nerva Archepelago
Longbow Warden 2: complete a player vs. player mission helping your side
Mook: get the Washed Up exploration badge in Port Oakes
Mook Capo: get with the Stone Cold badge
Nebula Elite Buckshot: reach level 10
Night Haunt: catch 50 ghosts in Fort Cerberus
Red Hand: catch 100 ghosts in Fort Cerberus
Wailer: get the Stone Keeper badge in St. Martial
To unlock the corresponding badge, complete the corresponding task in Supergroup mode:
Analyzer Base Defense: get damaged by 20,000,000 points
Cap Au Diable Beacon: get the five exploration badges in Cap Au Diable
Improved Igniter: get 100,000,000 influence outside of trading
Mega Moniter Plans: get 5 Supergroup badges
Mercy Island Beacon: get 6 exploration badges in Mercy Island
Nerva Beacon: get 5 exploration badges in Nerva Archepelago
Port Oakes Beacon: get 4 exploration badges in Port Oakes
Sharkhead Beacon: get 4 exploration badges in Sharkhead Isle
St. Martial Beacon: get 4 exploration badges in St. Martial