Vimes is bemused by the fact that the Low King is "crowned" by sitting on top of a sacred relic. But dwarfs regard things that are "low" to be better than things that are "high", so putting the Scone on his head would be treating it less respectfully than his perching on top of it.
The dwarfish superstition regarding the Scone is that the piece of Truth in it would flash white-hot if anybody touching it told a lie. The Low King is supposed to sit on it while passing judgement. Dwarfs like to keep their kings honest. And it would, of course, lead to a conflagration in the underwear department if he told an untruth...
Or to put in another way, any liar has his pants literally catch fire.
Although given how much metal most dwarves wear (and the implication from Unseen Academicals that much of it is against the skin), a king lying while sitting on the Scone would probably be encased in his own melting armor.
Also, while the Scone may not be the original, there have been plenty of books confirming how Belief is a fundamental force on the Discworld, and actually reshapes reality. So it is dwarves' belief as much as any metaphysical properties that make the Scone work.
Gavin and Carrot's fight with Wolfgang. Much is made of the fact that Carrot fights "by the rules"; but when Gavin jumps in, he's trying to fight fair, too. Which Gaspode notes before he jumps in to help.
Vimes, despite his avowed loathing for kings, hits it off fairly well with Rhys Rhysson. This makes more sense when you consider that it's really the concept of hereditary kingship that Vimes finds repugnant - by his reckoning, it's a recipe for breeding power-crazed unaccountable bastards - whereas the Low King is selected by consensus of the movers and shakers within dwarf society, not unlike how the Patrician himself gets chosen.
The Schmaltzberger dwarfs won't accept any dwarfs named Glodsson as candidates for Low Kingship. Why? In Witches Abroad, a footnote states that thousands of copies of a dwarf named Glod were created by a curse imposed upon a human king by a dyslexic god. The deep-downers probably wouldn't see their offspring as 'real' dwarfs.
Vimes and Sybil both privately hold the opinion that Carrot and Angua's relationship is under strain because of their concerns about not being able to have children. This is never brought up in either Carrot's or Angua's point of view sections, and it's made pretty clear that the problem actually stem from a completely unrelated source note Angua's anxieties that she will become more like her family and eventually turn on her non-werewolf friends, and that she can't be accepted by other species because she's neither a human nor a wolf. While it doesn't make much sense to apply this reasoning to Carrot and Angua, it actually makes complete sense for Vimes and Sybil to think like this, since it's a projection of their own situation: Vimes is afraid that they've missed their chance to have children by marrying in middle age; and Sybil is afraid because she's just discovered that she's pregnant with her first child, and is unsure of how Vimes will react, or the effect it will have on her health.
Vimes telling Detritus to blow the doors off the Baron's castle with the Piecemaker isn't just a Moment of Awesome: it also gives Vimes the satisfaction of papering over the memory of one of his previous minor embarrassments. Namely, the time in Guards! Guards! when he got carried away by Dirty Harry tropes and ordered Colon to shoot the lock off a door with a bow and arrow. Turns out all he'd needed to get that scene right was different Sergeant wielding a big enough bow.
Regarding the Scone's protective mechanism, the first part with the water-locks and boat could be Fridge Brilliance on the dwarfs' part. At first glance, it just sounds like another way of operating an elevator, but when you consider that vampires in Discworld (barring serious self-help training in the style of the de Magpyrs) have an aversion to crossing running water, you realize that it's not necessarily set up that way to keep out dwarf thieves.
Carrot's alarm when he asks Gaspode if the nearby villagers had killed the wolf they captured seems like an overreaction, given that it's unlikely they'd have used fire or silver against such an animal, so if (as he presumably fears) it was Angua in wolf-form, she'd potentially walk away unscathed. But if you recall the events of Men at Arms, Gaspode's report of the wolf's capture becomes a lot more alarming: it took Angua hours to revive when she was shot in that novel, and she didn't revert back to human shape while she appeared to be dead. So if the villagers had, say, riddled her with arrows until she dropped, they might well have burned her remains or, worse still, skinned her alive before she could wake up. (She'd look like a genuine dead wolf to them, and warm pelts are valuable in such a cold country.)
Vimes, watching the clacks at work, thinks that its inventors must be making money hand over fist. The poor fellows.