Binacle, the two-handed Pokémon from Kalos, can be found in the wild in Pokémon Go. Yes, Binacle can be shiny in Pokémon Go!
Binacle is mainly a Pokédex-filler Pokémon, as neither it nor its evolution have much use in PvE or PvP content. It costs 50 Binacle Candy to evolve it into a Barbaracle, so grab what you need and feel free to ditch the hand Pokémon.
As per old research by the now-defunct website The Silph Road (via Wayback Machine), the shiny rate for Pokémon on a regular day is approximately one in 500. Binacle is not a Pokémon who gets a "permaboost" (meaning that it's a rare spawn and thus gets a boosted shiny rate).
Not much, unfortunately. It appears to be random chance. Shiny Pokémon catch rates are set by developer Niantic, and they are typically only boosted during special events like Community Days or Safari Zones, or in Legendary Raids. There are no consumable items that boost shiny Pokémon rates.
LeekDuck keeps a list of currently available shiny Pokémon. It's a helpful visual guide that illustrates what all of the existing shiny Pokémon look like.
For more tips, check out Polygon's Pokémon Go guides.