Short answers, pulled from the story.
The name Avalokiteśvara translates to lord who gazed down at the world. It combines the prefix ava meaning down, lokita as a past participle of lok meaning to look or observe, and īśvara translating to lord or sovereign.
The name Avalokiteśvara first appeared in the Avatamsaka Sutra which predates the Lotus Sutra. Chapter 25 of the Lotus Sutra titled The Universal Gate of Bodhisattva Avalokiteśvara describes him as a compassionate bodhisattva who hears the cries of sentient beings.
Veneration of Avalokiteśvara continues to the present day in Sri Lanka where he is known as Natha-deva. The majority believe him to be the Buddha yet to come Bodhisattva Maitreya though iconography including an image of Amitābha Buddha on the front of the crown identifies Nātha as Avalokiteśvara.
Avalokiteśvara possesses an extraordinarily large number of forms and emanations including Thousand-armed Thousand-Eyed form with eyes on the palms of its countless hands. Other common depictions include Four-Armed Avalokitesvara holding a lotus and mala beads and Eleven-faced Avalokiteśvara has ten heads atop the Bodhisattva's head awakening living beings in the ten realms.
In Tibetan Buddhism the central mantra is the six-syllable phrase Om Mani Padme Hum. Recitation of this mantra while using prayer beads remains the most popular religious practice there alongside spinning prayer wheels clockwise containing numerous repetitions of this mantra.
In East Asian Buddhism the most popular form of Avalokiteśvara is the feminine white-robed Guanyin known as Southern Sea Avalokitesvara or White-Robed Avalokiteśvara. The Divine White-robed Avalokiteśvara Mantra dates to around the 11th century CE and she is worshiped widely in Taoism and Chinese folk religion alongside traditional Buddhist practices.