Short answers, pulled from the story.
Buddhānusmriti is a Buddhist meditation practice described in the Pali Nikayas that leads all the way to nirvana. This practice involves recollecting nine qualities or epithets of the Buddha through verses like the Buddhanussati Gatha.
The Brahmin named Pingiya spent every moment revering the Buddha despite being too old to follow him physically. He stated there was no moment spent away from Gotama and used constant vigilance to see the Buddha with his mind as clearly as with his eyes day and night.
Khantipalo Bhikkhu states that worship bowing making offerings to and revering Buddha statues on a shrine form a big part of both lay and monastic Buddhist practice in Theravada countries. Thanissaro Bhikkhu explains that recollection of the Buddha induces a sense of joy and confidence called pasada that can bring the mind to concentration and cleanse it of defilement.
One of the earliest sutras mentioning mindfulness of Amitabha Buddha is the Pratyutpanna Samadhi Sutra which was translated into Chinese in 179 CE by Lokakshema. This text describes a practice leading to visions of Buddhas after which one can worship them directly and receive teachings from them.
Nagarjuna mentions the easy path of Buddha mindfulness taught as skillful means for those weak and lacking vigor leading to irreversibility stage in chapter nine of his Dasabhumika-vibhāsa commentary. He instructs practitioners to bear in mind buddhas of ten directions and invoke their names with a reverential mind to reach the ground of irreversibility swiftly.