Short answers, pulled from the story.
The earliest stone record of Mahayana Buddhism dates to the year 180 CE. An inscription found in Mathura during the reign of King Huviska references the Buddha Amitabha and marks this early appearance.
The philosopher Nagarjuna founded the Madhyamaka school in the second century CE. His writings argue that all phenomena are empty of inherent existence or intrinsic nature known as svabhava.
Mahayana Buddhism spread along the overland and maritime Silk Roads connecting trade routes from India to China. Central Asian scholars like Lokakshema and Kumajiva transmitted teachings through these paths starting between 167 and 433 CE.
The arising of the Awakening Mind known as bodhicitta serves as the central defining feature of the Mahayana path. This wish to become a Buddha in order to help all beings marks the beginning of the bodhisattva spiritual journey.
Vajrayana emerged during the Gupta period around the third century CE. It continued into the Pala era spanning the eighth to twelfth centuries when it grew to dominate Indian Buddhism.