Short answers, pulled from the story.
Buddhists believe a properly rendered image is a hypostasis, an actual spiritual emanation of Gautama Buddha himself. This belief transforms the statue from a symbol into a vessel possessing supernatural qualities that allow devotees to communicate with the supernatural world.
The current range of postures and gestures evolved over the first millennium of the Buddhist era roughly spanning 500 BCE to 500 CE. A significant shift happened in the later 5th century CE at Sarnath in India where new Gupta period Buddhist images emerged.
Prince Paramanuchit Chinorot served as administrator of the Wat Pho royal temple in Bangkok starting in 1814 to standardize the appearance of Gautama Buddha. He described and represented 40 different postures of the Buddha in an illustrated treatise called Pathama Sambodhikatha at the request of King Rama III.
The Dīgha Nikāya lists 32 physical attributes of Gautama Buddha including legs like an antelope's and ankles like rounded shells. Other details include feet with level tread, long slender fingers, very long earlobes, and a protuberance on the top of his skull known as the ushnisha.
Touching the earth Bhūmiśparśa mudrā is by far the most commonly depicted gesture for Gautama Buddha. The right arm rests on the thigh with fingers pointing downwards while the left hand rests in the lap to represent Buddha subduing Māra the demon who tried to prevent him attaining enlightenment.