Short answers, pulled from the story.
The name Sarnath derives from the Sanskrit word Saranganath, which translates to 'Lord of the Deer' in English. This title refers to an ancient Buddhist legend where a male deer offered his life to save a female deer that a local king was aiming to kill.
Gautama Buddha delivered his first teaching at Sarnath circa 528 BCE when he was approximately 35 years old. The teaching itself is entitled Dhammacakkappavattana sutra and occurred within these grounds after attaining enlightenment.
Alexander Cunningham conducted the first systematic archaeological excavations at Sarnath during 1835 and 1836. By 1836, Cunningham had conclusively identified Sarnath as the location of the Buddha's first sermon.
Parts of the base and shaft of the pillar with its Schism Edict, lion capital, and remnants of the dharmachakra sculpture date to c. 241-233 BCE. These are the oldest and most important relics discovered at Sarnath thus far.
The modern Mulagandha Kuty Vihara opened to the public in 1931. Wealthy Hawaiian philanthropist Mary Robinson Foster provided much of the financial support for this project while Anagarika Dharmapala supervised its construction.