"Right hand man" and "first mate" aren't excluding terms. Rayleigh is literally introduced as the former first mate of Roger AND the right hand of the Pirate King. "The thread is about right hand and I'm arguing Zoro is not that but first mate" is an absurd statement, you can be both.
Again, your understanding of this is completely messed up. Out of Rayleigh and Zoro, the only one who has been explicitly addressed as the official first mate of his crew is Rayleigh; the only official role Zoro has been given is combatant. Saying Zoro is a first mate but Rayleigh isn't is factually wrong.
Regarding Kuma, and answering this too:
Zoro took Kuma's choice as a challenge and he faced him as such. Nowhere in that fight was addressed he was acting like the first mate but a warrior who is directly defied, which is his usual behavior; this isn't different from many other instances in which a character doesn't intervene in someone else's fight, and even then, Sanji interferes twice (kicking Kuma and offering his life). Of course a protective intention was involved, but I don't think any of them were truly aware of the threat they were facing nor I see Sanji following him as his first mate during the fight.
To summarize, he clearly is Luffy's right hand, right wing or however you want to address it, but I highly doubt Oda is ever making him the official first mate because that's a hierarchical role that doesn't really fit the crew's model based on interdependence (and, personally, feel like Oda would depict Zoro and Sanji in terms of Mohji and Cabaji if they ever had to decide a new captain; which won't happen anyways).
Anyways, this is getting off-topic.