Short answers, pulled from the story.
Clues to a figure's identity include its mudra (hand gesture), asana (sitting or standing posture), the objects it holds such as a book, thunderbolt, vase, jewel, or lotus, and any image in its crown. Context such as the direction the figure faces or the tradition that commissioned the work also helps identify it.
The five Dhyani Buddhas are abstract aspects of Buddhahood rather than gods or historical figures. Each must face a different direction (north, south, east, west, or center), and when painted each is a different color (blue, yellow, red, green, or white). Each also has a distinct mudra, symbol, animal vehicle, consort, and spiritual son.
The Laughing Buddha is Budai (Hotei in Japanese), an incarnation of the bodhisattva Maitreya, the future Buddha, not Gautama. His name means "Cloth Sack" for the bag of sweets he carries and gives to children. He is especially often confused with Gautama among Western audiences.
The Buddhas of Bamiyan, which included a statue of Vairocana, were blown up by the Taliban. China mourned their destruction and attempted to replace one of them with the world's tallest statue, named the Spring Temple Buddha.
Japanese Pure Land Buddhists believe Vairocana and the other Dhyani Buddhas are manifestations of Amitabha. Japanese Shingon Buddhists hold the reverse view, regarding Amitabha and the other Dhyani Buddhas as manifestations of Vairocana.
Avalokiteshvara is the bodhisattva of compassion and is considered to be incarnated in the Dalai Lama, who is a tulku and the most revered Tibetan Buddhist monk. Images of Avalokiteshvara can be mistaken for Gautama, making identification a challenge in Buddhist art.