Short answers, pulled from the story.
Alexander the Great crossed the Indus River and fought King Porus at the Battle of the Hydaspes in 326 BC. This military engagement marked the furthest eastern point of his conquests into the Indian subcontinent.
The famous Dharmaraksita is described in Pali sources as leading Greek or Yona Buddhist monks active in proselytism who founded the eponymous Dharmaguptaka school of Buddhism. These monks were part of a broader movement where Buddhist emissaries traveled to Greek lands in Asia and as far as the Mediterranean.
Alexander died on the 10th of June 323 BC, after which the Diadochi or successors founded their own kingdoms including the Seleucid Empire extending as far as India. General Seleucus set up this empire following the fragmentation of Alexander's territories.
Menander I became a Buddhist arhat at the end of his reign according to the Milinda Pañha. He established his capital in Sagala which was one of the centers of blossoming Buddhist culture and issued coins bearing the inscription Saviour King Menander in Greek.
Five monks from Gandhara traveled to Fusang country beyond sea introducing Buddhism there in 485 CE according to Liang Shu treatise. This event occurred during a period when Central Asian and East Asian monks maintained strong exchanges until around 10th century as indicated by Bezeklik Thousand Buddha Caves frescos.