Do we know the odds of acquiring Hidden Abilities in Raid Battles? Are there any ways of influencing those? Do they show in "normal" battles or only the rare "purple beam" ones? Also, can Synchronize affect the nature of Raid bosses?
The answer to all those is, unfortunately, no.
Purple beam seems to relate to the pool the pokemon is picked from, and nothing else (purple pool being the "rare" pool).
HAs rate seem to be around 10-20% chance, exact number is unknown but my personal experience definitely matches this (rip the HA araquanid i got...)
Syncronize seems ineffective (as the pokemon is generated when the Den spawns and is the same for all 4 people in case of multiplayer).
You can't influence anything in any way aside the weather, and if you do a method hasn't been found.
That said, those questions should probably be in the OI section SQSA rather than the competitive ones ;)