Is there a rare chance of a Pokemon being shiny? Are there any announced events for them? Are shiny Pokemon in the expert stages? etc.
There are Shiny Pokémon in the game but they are not chance encounters like in the main series titles. They are separate Pokémon with unique skills. Shiny Genesect was added as an Expert Stage, but the rest are typically Special Stages which you can download by checking in, and will occur on a repeating schedule until 2037.
Source
No. I am very certain there are no shiny Pokemon in shuffle. There haven't been in previous games (Such as Pokemon Link (Yes, a Pokemon game)) and the pokedex (Pokémon List) for these games do not seem to have the data to store shiny Pokemon. Source: Past experience