1. Don’t forget what happened to franky. He had an impressive showing against Luffy and then fought the gag character of CP9 Fukurou. Jack I also think is the only calamity with an awakened devil fruit which is why he’s able to take beatings like he does. He’ll dispatch dog and continue healing himself. He won’t be at full health, but he’ll be strong enough to give Yamato a hard diff fight.
2. I don’t think it’s too far. The amount of information we have on the beast pirates past is non existent. All we know is Jack sees Kaido as a father figure and King and Queen as his older brothers. This already shows a family attribute to his character which may connect to Yamato being kaidos biological child but Jack Kaidos chosen child. I think the grudge thing is over once he takes out dog. And seriously? Sanjis showing in WCI is a lot more impressive. Daifukus genie appeared much stronger than a gear 3 punch and Sanji over powered that. Don’t forget blocked a lightning bolt attack from big mom with Luffy. Any gear 3 move from Luffy barring red rock would be blocked pretty casually
Franky's debut? He was managing to take a troubled Luffy by surprise. Then he had a pretty even skirmish with the same Blueno who was just a step behind a Luffy who had his head in the game. So it checks out.
Jack's fruit or skill details, that's just speculation for now.
I also do think he's a little underrated though. Like people go by his disposition towards the other two but that's not necessarily about the strength gap. Kaido's even higher up the food chain but "father figure" is a stretch. That's more like a Newgate and Marco deal.
But honestly if you think Sanji "overpowered" the genie, that it correlates to Gear 3, and ultimately that base Sanji could block that when he couldn't handle a kick from Judge, I dunno what to tell you.