Trish-- you and I see Carrot almost completely differently. And part of the reason, I think, is because in your analysis of Carrot, you are using parts of the later books to bolster your view of Carrot. I don't want to get too far off the discussion of GG, but I think you're wrong about Carrot's nature as a character (and I'll eventually get around to showing you more precisely why.)
But as far as Carrot in this book-- the last two paragraphs of your post are, in my opinion, an almost total misreading of the character. Carrot not only takes the command to "throw the book at him" literally (like a good dwarf) and throws
The Laws and Ordinances of Ankh and Morkpork at Wonse, hitting him on the forehead and causing him to step backward and fall to his death. He thinks that's what he was ordered to do.
The hero of this scene is Vimes. Vimes doesn't intend to kill Wonse--though he'd come to the Palace to do so. Rather he acts like the Captain of the Watch--he upholds the law. But when Wonse resists, Vimes tells Carrot to "throw the book at him", meaning to arrest him. Unfortunately--(and lest you misunderstand Carrot's literalmindness, Pratchett writes
"
Vimes remembered too late.
Dwarfs have trouble with metaphors.
They also have very good aim." (emphais added)
In fact, when all the Watch members are looking at Wonse's dead body and Colon comments that he's been "Killed by a wossname. A metaphor," Carrot is quite concerned and asks the Captain, "That was right, wasn't it, sir?" said Carrot anxiously. "You said to--" and Vimes has to calm him down.
Carrot was following his understanding of orders just as much here as he was when he arrested the Head of the Thieves Guild, and almost arrested the Patrician.
In this book, Carrot - a 16+ year old boy who has been raised in a mine by dwarfs and come to the city where he has his first real contact with human beings (except for the mailman who gives him the Protector) is Pratchett's perfectly naive, dwarf-raised and therefore absolutely literal-minded comic character.
The only "heroic" things he does is jump off roofs (and pull the other watch members to safety). Even his "great battle in the dwarf bar" only proves that he is incredibly naive and incredibly strong. It accomplishes nothing for the city. It sets up the rest of the Watch as they are at the beginning of the book--a bunch of drunken misfits. But it's a wonderful comic scene!
Carrot doesn't even know how to use the sword, which his father gave him just before he left home. It is Vimes, at the critical point, when Wonse attempts to kill the Patrician with the "royal" sword, who finds himself (for reasons he doesn't quite understand) using Carrot's sword to block Wonse's blow. And that's how the whole question of whether this is a special sword - belonging to the kings--comes up. Because Vimes has to "do it by the book" and his blocking stroke cuts the fake sword in two.
In this book especially Carrot is nothing but a fairly minor comic figure, the hick from the sticks who has all the literal-mindedness of the dwarfs who raised him. Vimes, on the other hand, discovers the first evidence (albeit in a drunken stupor) of the dragon, finds the one person (Lady Sybil) in the city who can educate him about dragons, tries to follow orders while trying to solve the crime and find the dragon. Incidentally he wins the fair lady, aids Erroll in his quest to transform himself, and saves Ankh-Morpork.