Short answers, pulled from the story.
Hiero II of Syracuse was proclaimed king after a victory in 265 BC when fighting the Mamertines prior to the formal start of the First Punic War. He had been appointed commander-in-chief by the Syracusan army and citizens in 275 BC before seizing total power between 275 and 271 BC.
Hiero II of Syracuse defeated the Mamertines, a body of mercenaries from Campania who had seized Messina as a base of operations, in a pitched battle near Mylae along the Longanus river. Carthaginian interference prevented him from capturing Messina itself but he gained control over much of the territory.
In 263 BC Hiero II of Syracuse concluded a treaty with Rome that allowed him to rule over the south-east of Sicily and the eastern coast as far as Tauromenium. This diplomatic shift marked a turning point in regional politics during the conflict and established boundaries that would define Syracuse's territory for decades to come.
Archimedes worked with Hiero II of Syracuse to construct engines designed to protect the city from naval attacks and ground assaults during the siege of Syracuse by the Romans. These machines helped delay Roman conquest despite overwhelming odds and transformed Syracuse into one of the most heavily fortified cities in the Mediterranean world.
Hiero II of Syracuse suspected he was being cheated by a goldsmith regarding a votive crown and asked Archimedes to verify if all the gold had been used as agreed upon. Archimedes discovered the principle of displacement needed to measure density while noticing the rise in water level when getting in a bath tub and successfully detected the fraud.