Kuzco is 17. You don't have to be a royal teenager to think It's All About Me. Plus, Yzma herself said that she "practically raised him." The apple doesn't fall far from the tree indeed.
Why doesn't Kuzco have a food taster? It's because he thinks the whole world adores him. The thought that someone might dislike him, let alone try to kill him, never even crossed his mind.
Pretty much confirmed when Pacha tries to warn him that Yzma and Kronk are out to kill him:
Kuzco: Kill me? Their entire world revolves around me!
When Yzma's goons all get turned into a random assortment of animals, none of them seem nearly as alarmed as you'd expect... save one guy, who asks to be sent home early on the grounds that he is now a cow (Yzma allows it). Why is he the only one showing any concern? Well, despite what pop culture might have you believe, cows are exclusively female. Poor guy didn't just have his species changed, his biological sex was changed as well, which is disturbing and unexpected even by the standards of someone used to Yzma's transformation antics.
Yzma asks Kronk "Why do we even have that lever?" Well, if Kronk was just a touch more on the ball, it would be a great Covert Distress Code - if the wrong person is trying to get into the lab, they get dropped into the Crocodile Pen. Too bad that Kuzco and Pacha just get right past the crocodiles with little problem, so this happens:
Kuzco: Okay, why does she even have that lever?
In a subtle bit of the opening scene, Kuzco looks back and slips- that's why he falls in the mud. It's an amusing metaphor for what he tries to do before undergoing Character Development.
Some Fridge Humour: Yzma's original plan probably would have worked so long as the potions weren't mixed up, but she decided not to go through with it because she wanted to save on postage. The entire movie only happened because she was too cheap to buy a couple of stamps.
How far away does she even live from the emperor that she'd need to spend that much on postage? There's cheap, and then there's Yzma!note And that's not even considering the presumption that she'll expect Kronk to do all the dirty work and mail the box to her instead of doing it herself and SMASHING HIM WITH A HAMMER then and there.
As Yzma smashes Kuzco busts in a rage, she mentions that she practically raised him. Taking into account Kuzco's parents are probably dead if he's ruling, this could very well be true, and would certainly explain how he grew up to be such a spoiled brat.
As noted earlier with the possible dead parents thing, Yzma might just be the closest thing Kuzco has to a mother. Which could be why he's so broken when he finds out that she's dead set on killing him.
Almost every time that Yzma's potions are used in the movie, they work instantly…except during Kuzco's transformation scene. Which is weird, until you consider the fact that not only was the potion watered down by the wine initially, but then was further watered down in Kronk's hasty attempt to make sure Kuzco got a wine with potion. Thus, not only is the slower transformation justified, but it would almost be odder if it didn't happen. In fact, notice that the three clouds that are produced from the cups are considerably smaller than the first cloud.
And, of course, Yzma raised Kuzco. She probably passed her accidental self-sabotage trait onto him.
In the beginning of the film, Yzma orders Kronk to kill Kuzco, but he has an attack of conscience and can't go through with it. Later, she gives Kronk a dagger and tells him to kill Kuzco and Pacha. Why does she let Kronk, who has let her down before, try to kill them instead of just doing it herself?
She knows that Kronk failed to kill Kuzco the first time, but doesn't know why. She probably attributes his failure to his general stupidity and incompetence. Now that she's watching him, she can make sure he doesn't try something silly that's doomed to fail. "Take this dagger and stab them to death" is not a hard instruction to follow - at least if you don't have an interfering conscience.
Yzma is a thin, old lady. Kronk is a muscular guy who's much younger. Kronk has a much better chance of overpowering Kuzco and Pacha.
In case they're caught by someone who happens to know/discover that the llama is Kuzco and that Pacha is innocent, it'll be easier for her to put the blame on Kronk. She can even say that she would've tried to save them if she weren't an old lady who wouldn't stand a chance against him. People would probably trust the emperor's adviser more than her random Dumb Muscle minion. Even if she does get punished, it might be more lenient because she wasn't Kuzco's killer.
Why is everyone so quick to accept Yzma as a new ruler? It's possible that at least some of them adored Kuzco out of obligation, not because they genuinely thought he was the best person ever.
Not only that, she's been at the top level of government for a long time. She's been Kuzco's adviser/parent since he was born, and presumably she was his parents' adviser when they were still alive. Why wouldn't they trust the old, experienced woman who's been working just below the emperor for ages?
When Yzma and Kronk were chasing Kuzco and Pacha, they were struck by a lightning bolt and fell down a ravine. Even though they shouldn't make it to the laboratory before these two because of this case of bad luck, they did so anyway. Which is immediately lampshaded complete with a graph. The scene is already funny enough on its own, but it arguably gets even better if you think of the two as having fallen into a hole. In other words, they fell into a Plot Hole.
People say that Kronk cooking dessert for the assassination attempt dinner is Comically Missing the Point, but it would have been pretty damn suspicious if Kuzco had died at a dinner that nobody bothered to finish cooking! (Of course, there's no telling whether or not Kronk thought of it that way.)
It might not be intentional, but Pacha's Sickening Sweethearts act appears to be his Chekhov's Skill: first he uses it to make his kids go to bed when he says that he and his wife will just be outside "saying how much they love each other." Later, he uses it so he and Cuzco can pose and a newlywed couple at the diner.
Kuzco is already emperor as a teenager, with his parents nowhere to be seen. Considering that Yzma attempts to murder Kuzco to seize power, it is entirely possible that she did the same to his parents while he was still a little child...
Yzma claims that only herself, Kronk, and Kuzco know that she was fired. However there was an aide writing down what Kuzco was saying and even gave her a note with all the different ways she was fired.
She probably just considers the aide too insignificant to worry about, especially since the word of the emperor's adviser would definitely be taken more seriously than the word of a random aide.
Or, going by the previous Fridge Brilliance, maybe the aide liked Kuzco out of obligation and would be rather indifferent if he just disappeared randomly.
Or, since the aide was only present while Kuzco was rhyming off euphemisms — not a second earlier, and not a second later — he may not have gotten enough context, or even paid the slightest bit of attention to what he was writing, to actually have any clue what was going on.