Heavens to Murgatroyd! Miss a day or so here, and there's a whole volume to read and try to condense to reply to various points people have made.
First, as to the kingship-- Terry is, I think, going to use Carrot as a special kind of rightful king and is perhaps setting it up that way in this book. In real world terms, the heritage of Carrot as King makes no biological sense. But Pratchett's going to develop him into a King in the next book, before Terry either gets bored or "painted into a corner" with the character of Carrot. In my opinion, there is a slow but steady decline in the interest and likability (by the reader and also by the author) in the character of Carrot. He's too perfect most of the time.
But in this book--Carrot is the comic character
par excellence -- he's the BIG, DUMB HICK FROM THE COUNTRY, a standard comic type. (Think Gomer Pyle) And as such, he's funny in the kinds of mistakes he makes (or almost makes) because of his naivete and his 'dwarfish simple-minded-focused-no metaphors' nature. (He arrests the Head of the Guild of Thieves, lives in a bawdy house, etc., etc. ,etc. )
But I really doubt that Terry ever intended this book to be a stand alone. I think that all the things about the sword, the birthmark and HIS ABILITY TO GET PEOPLE TO DO WHAT HE WANTS - are foreshadowing of the kingly abilities that will come out in the next book.
Sybil, I think, is big because of Terry's original intention to use her as another comic character--the "mad old lady who raises swamp dragons". But Sybil, even in this book, has signs of her noble blood. If you compare the passage of Vimes drunken ramblings about his love for the city--so big, so overwhelming, so lovable--and then she'll kick you in the teeth (paraphrased) with his later description of Sybil as having a heart like the city, you'll see that their relationship (and possibly their dragon cohorts relationship) are the real million to one chances that come true in Discworld. There's a great deal more I want to say about Sybil, but not in this post.
And, as Tina said of her man, I too am living proof that million to one chances "just might work."