Basically, there's a very long-running time travel plot going on. It turns out the Queen has a rather powerful bit of magic going on (there's a REASON this kingdom's got a matriarchal line of descent...) in that she apparently time travel magic and can propel a person's consciousness backwards in time. Originally, she used this to navigate a previous succession crisis and take the throne.
She used it in a Bad End Future on the King and herself. In that future, Souma didn't get made king, just the prime minister, and while he did his best, he apparently didn't have quite enough direct power to deal with the corrupt parts of the nobility fast enough, the Corrupt Nobles tried to do some sneaky stuff (which Souma and the parts of the government he was working with didn't realize and went along with in Bad Timeline), outside threats took advantage (the current invasion), etc etc etc and the end result was the Kingdom ended up destroyed.
That's the ACTUAL reason the King was so quick to go 'y'know what? I abdicate. This guy's king now. Also, I engaged him to my daughter!'. He'd already gotten a good grasp of Souma's character in the future, and saw that the relationship with his daughter might have worked, since she fell in love with him in the Bad Future naturally without an engagement, but that got spoiled by the whole 'Things going horribly wrong and everyone died'.
(The fact this gives him an excuse to hang out in his modest home, relax, and be lovey-dovey with his incredibly hot wife is just a bonus, really).
The reason Carmine did his crazy-ass rebellion is that he's STILL ultra-loyal to the King and Kingdom, as well as ultimately Souma. King Albert took him aside and quietly told him what happened in the original time-line, and since Carmine is completely loyal to the King, he 'rebelled'. He's basically purposely acting as a lightning rod for the worst elements of the Kingdom (the corrupt nobles) to seize their assets and the nobles to hand over to Souma. Plus, he's drawing in all the local disreputable mercenary forces available (IE, the guys willing to take the Corrupt Nobles' money and also the ones most likely to turn to banditry or be used by the neighboring hostile kingdom as reinforcements) and setting them up to be butchered. If you notice from the previous chapter, his well-trained, well-armored soldiers were standing back while the nobles' mercenary and personal guard were getting mauled attacking the keep's walls with basically swords and not much else.
Duke Wolter guessed half the plan back in Chapter 11, by the by, so it's not actually that far out of left field. What she misjudged is just how loyal Carmine is to the Kingdom; she just thought he was planning to become the power-behind-the-throne.
Castor, the Air Force Duke, is mostly just backing his best friend Carmine up and hoping like hell he knows what-the-fuck he's doing. Wolter's already decided to openly side with Souma after her grand-daughter Juna gave him a thumbs up, character and competency wise.
And yes. This means that the Man Behind the Man, who's set up the entire current plot, is the goofy, ineffectual seeming King Albert monkey-wrenching the piles of sneaky plots infesting his Kingdom and the neighboring Kingdom's antics via abdicating and putting Souma in charge.
Apparently, after the rebellion and invasion are put down, the fallout is that Carmine is 'executed' (and a Mysteriously Large Lionish Beastman With A Mask becomes head of the Kingdom's special ops) and Castor's daughter Carla is sentenced to slavery in the castle as a maid (which is his punishment for rebelling) and she ends up with direct orders from Souma himself to kill him if he ever slips into becoming a tyrant instead of a just king. Which might actually be a geas or the like, not just orders, so she's basically turned into his Sword of Damocles.