Usually when a character's imbuing their attacks with Conqueror's haki, it's highlighted. I don't think Jimbei has it, seems more like very proficient armament haki. Kinda hard to differentiate at times since black lightning isn't even unique to haki, let alone Conqueror's haki.
As for the left-hand man thing, Zoro and Sanji were referred to as the Pirate King's Wings, only official pairing I've heard and sadly it wasn't Jimbei but Sanji involved. Jimbei's still significant, he's even rocking a higher bounty but Sanji's solidified his spot as a man worthy of being called one of the Wings of the Pirate King per Robin, a long-time crewmember. Jimbei proved himself during the Summit War Saga and Whole Cake Island but he's not one of the Wings. Nothing wrong with that, there's only two.
In regards to Zoro, it's a question of whether he imbued his Ashura with Conqueror's haki or he simply emitted it the usual way like unconscious Luffy or Kid did whenever Kaido took note. I think it's the latter to be honest and he scarred Kaido without it then upgraded everything when he started applying Conqueror's to his attacks against King (plus with the King of Hell naming to differentiate it and make it clear he's imbuing).
So, the way I see it, either Zoro used it first on his Ashura followed by Luffy first figuring it out then later Luffy mastering it and sky-splitting or Zoro first used it against King where he became proficient enough to have a mode/number of attacks based around it.
Overall, Jimbei's not the left-hand man. I don't recall anyone using that title on the Straw Hat crew whereas Monster Trio and Wings are actual terms used by characters in the series and the author. Jimbei wasn't given one but atleast he has his bounty. Luffy's the most proficient at Conqueror's at this point, figuring out Kaido was using it on Thunder Bagua then replicating that imbuement followed by refining it later and splitting the sky. Zoro comes next for using it to end his fight and Jimbei hasn't been confirmed to have regular Conqueror's, let alone imbuement. But he's still cool.