You can legally take eurasian collared dove and quail with an airgun in CA, but not other species of dove.
As for airguns in a shotgun only areas, I suspect the answer is yes. But that is just a guess and you should call DFW to have that question answered. The shotgun only designation is meant to prevent rimfire or center fire firearms from being used in those areas for safety reasons. Airguns are not considered firearms in CA so it seems reasonable they would be permitted.....but CA isn’t known for its reasonable hunting laws.
R