Short answers, pulled from the story.
Greco-Buddhism is the fusion of Hellenistic Greek and Buddhist culture that developed in Gandhara, in present-day Pakistan and north-east Afghanistan, between the 4th century BC and the 5th century AD. Most scholars find its clearest evidence in art rather than in religious belief itself.
Greco-Buddhism developed between the 4th century BC and the 5th century AD. Contact between Greek and Buddhist culture began after Alexander the Great's campaigns and continued through the Greco-Bactrian Kingdom, the Indo-Greek Kingdoms, and the Kushan Empire.
The Indo-Greek king Menander I, who reigned around 160-135 BC, may have converted to Buddhism and supported its spread. According to the Milinda Panha, he became a Buddhist arhat by the end of his reign, and Plutarch wrote that his relics were divided and enshrined.
Emperor Ashoka converted to Buddhism after the Kalinga War and recorded his efforts to spread Dharma in the Edicts of Ashoka, some carved in Greek and Aramaic. He claimed to have converted Greek populations within his realm and sent Buddhist emissaries as far as the Greek kings Antiochos, Ptolemy, Antigonos, Magas, and Alexander.
Greco-Buddhist art developed primarily in Gandhara, a region centered on present-day Pakistan and north-east Afghanistan. Sites such as Hadda in Afghanistan and Sirkap near Taxila have yielded sculpture that combines Buddhist subject matter with Hellenistic style.
The term Greco-Buddhism is controversial because most scholars do not find a noticeable Greek influence on Gandharan Buddhism beyond the artistic realm. Olga Kubica has called the term inadequate, and Johanna Hanink has argued that the related concept of Greco-Buddhist art reflects a colonial-era reluctance to credit native artists.