Are the stats from Zero Beta's passive skills shared with Alpha?

And viceversa. Like for example: "Solid Body: Enhance Beta’s body, permanently increasing her defense and magic defense by 2000, resistance to statuses and elements by 50%, and Stance by 100%. (required level: 110) (max level: 20)"

would work for Aplha and Beta 100% of the time while on tag mode.

Not really, only the bonuses that can be shared is their Rapid Time skill, and that's only in Tag Mode.