Whose bright idea was it to lock up live Free Magic nasties in the basement, anyway?
They may not have been invited in the first place, they may have snuck in. Plus, not all of them can be dealt with by Clayr; the Stilken, for example, can only be killed with Free Magic, something no self-respecting Clayr is going to use even for a good cause, but it can be sealed using Charter Magic. The others might have been kept around as research subjects, then forgotten about. They should really either stop doing that, start putting them in slightly less stupid places, or start taking inventory every couple of decades to make sure nothing unpleasant gets out, but where would the fun be in that?
It's also stated that by living so much in the future they tend to forget the present. There may be a bit of 'Bystander Syndrome' mentality if they don't See themselves doing something about it.
The delay for categorising a new aquisition to the library is 50 years minimum.
Sabriel is killed, but when she enters Death she is greeted by the spirits of her ancestors, who tell her she cannot go with them until there is another to carry on the family line. What's Lirael besides someone to carry on the family line? Sabriel's own kids ended up being irrelevant to that bloodline anyways.
Well, the two obvious reasons are: A.)Garth Nix hadn't thought up Lirael yet B.) Lirael was barely a newborn baby at that time.
This falls into the category of "Blindingly obvious once you realize it," but the reason Sam has so much trouble trying to read The Book of the Dead isn't just PTSD regarding anything to do with Death, it's that the Book can only be opened by someone with a natural talent for Free Magic and necromancy. Sam doesn't have one.
Actually, Sam does have some talent for it, as he is capable of crossing into death, but he just wasn't meant to be the Abhorsen-in-waiting, and the book knows it.
Having just re-read the series, Sam does indeed quite plainly have the raw talent to be an Abhorsen. It's his personality that is much less suited for the job, being more in line with his real gift, that of being a Wallmaker. I'd figure that if something had happened to Lirael he could have been an Abhorsen (though I doubt he'd be a particularly good one); with her there as a much better candidate to take over that side of the family business, however, the paraphernalia (Book, bells, etc.) naturally gravitates to her and Sam is left free to pursue his true destiny.
Although this could also fall into "Blindingly obvious", Lirael reading about the panpipes in The Book of Remembrance and Forgetting. "Though generally the instrument of a Remembrancer, the pipes are not infrequently used by Abhorsens-in-Waiting, till they succeed to their bells." Which is precisely what happens to Lirael at the end of the book, when she discovers she is an Abhorsen rather than a Clayr — she succeeds to the bells. And, given what we know about Remembrancers, it seems likely that this has happened before.
The Charter bloodlines are the answer to everything. How does Kerrigor have so much power? Power enough to actually break Kibeth and Saraneth when he needs to, power to set about breaking the Charter, and nearly succeeding? Because he himself is one of the Great Charters — the Royal bloodline. He is tied to the Charter itself, so if anyone can break it, it'll be one of the family. (In which case, Chlorr's story will be extremely interesting...)
And on that note, why did he try to use the blood of his family and the Abhorsens to break the Great Charter Stones? Why not that of the Clayr, who are far more numerous and would probably be at least a bit easier than waiting for two hundred years for a vulnerable Abhorsen to come along? Because the Clayr can see the future, and are especially sharp on Seeing that which pertains, personally, to them. Probably the only point that the Clayr's Sight would fail them would be when every other bloodline had been used to break the Great Stones — which would leave only about two or three left.
Honestly, once the Clayr Saw what Kerrigor was up to, they probably just hunkered down in the Glacier, a fortress even he couldn't conquer until he'd already destroyed all else, and waited him out except on occasions they Saw something particular they needed to do (like sending Sanar and Ryelle to advise Sabriel and Touchstone).
The Dead themselves. The protagonists are always talking about innocent people being killed and having their spirits enslaved and hoping like heck it doesn't happen to them, but it's all too easy to forget that a lot of the baddies that are doing things of their own relatively free will, all the Greater Dead, used to be human as well.