In Last Argument of Kings we learned that Bayaz happily employs Eaters of his own, that he considers the First Law to be meaningless and he did in fact kill Juvens. So if he has no grounds for looking down on Khalul, why does he want to destroy him so badly?
I always thought it was because Khalul wants to kill HIM for killing Juvens. So he has to kill Khalul first. And also because he really hated Khalul for reasons going way back. You can say this for Bayaz, he can hold a grudge and he never lacked in arrogance. I don't really think Bayaz needs a reason though.
That was probably the original case, but I think the books are pretty clear that it's basically a pointless, murderous personal feud between the two of them by the time of the books. That's part of the general subversion of fantasy stereotypes that Abercrombie does in the First Law Trilogy, with the "great quest" and/or "epic conflict" being nothing more than the personal enmity between two powerful beings.
Why did Tul Duru get himself killed?
Supernatural origins or not, Logen's persona as the Bloody Nine is very similar to a berzerker rage, which is not unknown in societies like like the Northern tribes. Crucially, this also means people generally know how to recognize and deal with berzerkers- namely, stay the hell away from them. So, why did Tul Duru "Thunderhead," one of the Named Men who had been travelling with Logen for some time, fail to recognize Logen was in his angry place and insist on walking up to him?
It actually doesn't seem as if the Northmen do know what a berserker rage is. The way they treat Crummock-i-Phail suggests that they prefer to do war professionally, and that people who start killing indiscriminately in that way make them nervous. No one seems to actually understand that the Bloody Nine and Logen are two separate personalities; the fact that he doesn't act like the Bloody Nine normally just cements in their heads the knowledge that he's a crazy, unpredictable fucker.
Whether berserker rage is common in the North or not, everyone knows that Logen can be a totally unhinged lunatic. On top of that, Tul Duru has been fighting with Logen long enough that the Bloody Nine shouldn't have come as a surprise.
It's worth noting at this point in the books the northmen have been defending a siege for seven days, which throws up several explainations. 1) Logan had fought for seven days without the Bloody Nine taking over once. Why would Tul assume that at this point, the Bloody Nine take would take over, when it had not happened so far in the whole battle? 2) Logen had gone down in the middle of a pitched Melee. If Tul had not pulled him out, he'd be dead. 3) Tul had been fighting for seven days. Even if he should have noticed that Logen was about to go crazy (as he'd see it) there is a damn good chance that he was acting on instinct.
It's also pretty clear that he thought Logen was about to die; he really just made a serious misdiagnosis.
There's a scene in LAoK where Crummock says to Logen that the two of them have a habit of being "none too particular" about who they kill when the fighting gets rough. That's probably about as far as the Northmen's knowledge of berserker rage goes.
I think the truth is that the Northmen do know what a Berserker Rage is...and they know that this kind of stuff ain't no Berserker Rage. Similarities aside, the simple fact is that Logen was not suffering from that and if Crummock-i-Phail is an example of what it is like, then consider that even he thinks that Logen is seriously f*cked up in the head. And I think from Black Dows point of view that wasn't just a berserker going nuts; from what he saw Logen looked at his victims and not only killed them anyway, he let the enemy take the fall for it. Even if it was berserker rage, Logen was apparently calculating and cold blooded after the fact and obviously isn't going to let it get to him that he murdered kids and his own ally. A Berserker Black Dow could let slide; the Logen he thought he saw is a bastard even he can't stomach, and definitely can't tolerate being made King since he'd probably butcher them all. As far as Tul is concerned, this still applies- even Berserkers don't do what Logen did('nt).
Why did Goyle cover up Quai's murder?
Forgive me if I missed it, but I cannot recall any point in the trilogy we are given any inkling that Goyle is so much as aware of Tolomei, let alone any indication that he could possibly be in cahoots with her, or even any reason that he might have to end Glokta's investigation. So what's up with that "dogs!" scene?
Unless Abercrombie brings him back as a protagonist in a later book, there probably wasn't meant to be anything sinister about it. Goyle is just all-around stupid and corrupt, and hostile to Glokta.
Goyle was, like most higher-ups, being paid off by Valint and Balk. Bayaz knew of Glokta's interference from the moment he asked Severard to investigate the bank, and Goyle's strange behavior when talking to Glokta is explained by the same fear exhibited by everyone who works for Bayaz.