Religion. Even if it's worshiping the Emperor. Especially if it's worshiping the Emperor.
Don't tell the Emperor that you have costed him a Emperor-class Battleship when other options were available. Just don't. That said, he'll forgive such an act if it actually accomplishes something meaningful.
Also don't ask him about the Traitor Legions. His anger at this line of thought makes the loss of an Emperor-class Battleship look tame by comparison.
Butt Monkey: The Emperor's caretaker and Custodes Captain-General. Half of the time, he's getting insulted by the Emperor for being a moron, while his own fellow Custodes mock him despite him being Captain-General, which they didn't even know about until Fyodor invaded the Palace
Card-Carrying Moron: The Master of the Administratum (third High Lord from the left) isn't just a useless piece of shit, he aspires to be a useless piece of shit.
Cerebus Syndrome: The first few episodes were dedicated to the Emperor caustically commenting on the state of the Imperium/thinly veiled Take Thats at the 40K canon. It isn't until the Emperor decrees that the Inquisition be disbanded that things start to get somewhat serious.
Cuteness Proximity: The Emperor loves having a Centurion seated on his lap because he thinks it's adorable. Inverted when he sees the Dreadknight, which He finds absolutely horrible.
Dude, Where's My Respect?: The other Custodes have no respect for their Captain-General, or as they call him, Little Kitten. Hell, until episode 11, they didn't even seem to know which Custodes their Captain-General WAS. Overall they continue to treat him as competition at best.
In a similar vein, the Emperor consistently orders the Custodes to put some damn armour on, which they openly and flagrantly ignore.
Easter Egg: The binary text in the "ASK ME QUESTIONS!" mini-episode reads: "I can't wait for Cypher to get to Mars so he can execute the plan." and "I think the Ultramarines are coming around with Magnus soon." The third message started out "As long as the questions don't scream about Baneblades" but was cut off by the Emperor.
Embarrassing Nickname: The Custodes in The Throne Room is referred to as "Little Kitten" by the other Custodes.
Establishing Character Moment: The first words out of Emperor's mouth- er, speakers, are "About fucking time. I have so many things to complain about."
Everyone Has Standards: The High Lords of Terra outlawed saying the name of the Old-Folks-Home Planet and don't react well when Fyodor mentions going there to find new High Lords of Terra after killing the present ones.
Evil Is Petty: Tzeentch comes across less as the Eldritch existing-in-the-past-present-and-future mastermind, and more as a petty little bully.
Exact Words: While Ecclesiarch Decius reluctantly agreed to let the Inquisition conduct their investigation in Holy Terra about the disbandment message, he told Karamazov not to spill the blood of its population. Karamazov agreed, and promptly told his men to burn them instead.
Fantastic Racism: Surprisingly averted with the Emperor. He reveals he doesn't mind living in peace with the other alien races, he just prefer they accept his idea of peace, mainly the aliens under his rule and not the aliens idea of peace (eg. Eldar's idea of peace is keeping humans like zoo animals.) Furthermore, he thinks the current Imperium's goal of trying to exterminate all aliens in the galaxy is not only impossible but also impractical.
Foreshadowing: Our Custodes friend, the Emperor's caretaker, claims to have been the one to meet Alicia Dominica during Goge Vandire's Reign of Blood. In canon, the Custodes who met Dominica was of the rank Centurion, second only to the Captain-General. This is pretty much the only clue we get to our Custodes friend being of a much higher rank than his Butt Monkey status makes him look.
Fourth Wall Mail Slot: Episode 12.5, while technically it is an in-universe Q&A with the Imperial citizens (and a few non-Imperials), there a few moments of breaking the fourth wall. Also, one of the notable things to come out of this is that there way too many to ask the same question: What does the Emperor thinks of Tau?. Since he doesn't know them, he ask the Custodes, who answers that they are insignificant, to what the Emperor counters that with how many citizens asked about them, they cannot be insignificant.
Freudian Slip: The Emperor's webway may or may not have been an excuse to score some Eldar hookers. Similarly conversations about the Tau may or may not be a place for our Custodes friend to make penis innuendos.
Fun with Subtitles: Occasionally, subtitles have their own gags added to them to emphasize certain things. Helps a lot when the most talkative person in the series doesn't have any ability to emphasize naturally.
Grumpy Old Man: The High Lords of Terra are basically what 1d4chan imagines them to be: a bunch of old, useless farts who make things worse through their idiocy.
The Emperor acts like a grumpy dad, equating the fall of nine of his primarchs to Chaos with "rebellious teenage phases".
He-Man Woman Hater: The Emperor recalls the end of the Reign of Blood less favorably and explains that he didn't make female Astartes because he thinks girls are yucky.
Heroic Sacrifice: The series takes a Broad Strokes approach to the three versions of Ollanius Pius story: all three of them happened! The lone Guardsman AND the Space Marine Terminator AND the Custodes had all thrown themselves between Horus and the Emperor, in that order. The Big E thinks that the latter two did it so that they could save face from being upstaged by an ordinary human.
Holier Than Thou: Karamazov seems to believe he's much holier than everyone so therefore his actions are justified despite the fact the Ecclesiarch and the Custodes, the Emperor's personal servants, call out on his actions.
Emperor: I DESTROYED ALL ASSHOLE RELIGIONS THAT EXISTED ON TERRA. DO YOU WANT TO KNOW WHY?
Custodes: Because you are the one true God, my lord?
EmperorWRONG. IT IS BECAUSE RELIGION IS STUPID, SUPERSTITIOUS BRAINWASHING CRAP THAT MAKES YOU AN ASSHOLE. THIS IS WHY I SPECIFICALLY SAID WHEN DESIGNING THE IMPERIAL TRUTH, THAT EQUALITY, SCIENCE AND GALAXY CONQUEST IS THE WAY TO GO. AND RELIGION NEEDS TO BE THROWN OUT A WINDOW.
A God I Am Not: As with the canon, the Emperor has major gripes with being called a god, and because of this disbands the Inquisition and Ecclesiarchy.
Inconvenient Itch: In addition of feeling endless torment of having his soul torn apart bit by bit by feeding the Astronomican, the Emperor also has an itch on his nose, despite not having the nose anymore, courtesy of Nurgle.
I Shall Taunt You: Tzeentch utilizes cheap insults and clever remarks to slowly win Magnus over to his side.
Meanwhile Scene: When our Custodes friend is about to explain the Inquisition to the Emperor, the video cuts to the Imperial Guard in middle of several warp storms.
God-Emperor: I THINK I SHAT OUT AT LEAST 5 WARP STORMS DURING THAT PRESENTATION.
Happens again in Episode 12 when the Custodes, eager to change the subject, mentions the Traitor Legions. Then the episode cuts to Mars, with some tech-priests discussing a newly discovered STC-fragment.
Most Annoying Sound: In-universe; helmet-mounted lingua-technis mouthpieces sound like bad dial-up modems, and the Emperor psychically punches anyone using them.
Only Sane Man: The Ecclesiarch is the only High Lord who isn't a frothing idiot.
Parental Favoritism: The Emperor badmouths all his sons, loyal and traitor alike. Except Sanguinius.
Perpetual Frowner: In Episode 7, Emps demands that he always have someone close-by to "tell me stupid shit I can complain about". He also manages to pull this trope off in the literal sense despite being a skull, which has to count for something.
Pet the Dog: The Emperor is actually impressed by people who are willing to sacrifice themselves meaningfully. He badmouths all of his Primarch sons except Sanguinius, he compliments Ollanius Pius for standing between him and Horus after Horus nearly slew him (and badmouths the Custodes and the Terminator who also threw themselves to be destroyed by Horus because they didn't want to be upstaged by some plain human), and although at first he dismisses the Astral Knights, upon hearing that they crashed a Battlebarge into a World Engine, cracking its void shield enough to get down on the planet, and died fighting and successfully crippling the enemy ship, the Emperor immediately revises his opinion of them and claims that he totally created them.
Holy shit, a giant skeleton space robot planet that shoots molecule-breaking beams and destroys everything it spots... I want one!
Reasonable Authority Figure: Among the High Lords, the Lord Commander Militant of the Imperial Guard tries to be this to a very minor extent. He tries to impede the other High Lords from weakening the Imperial Guard, by keeping laxatives and air legal.
Also, the Ecclesiarch, even after having his organization disbanded by the Emperor, tries to prevent Karamazov from killing all of the Custodes.
Bitter and caustic as he may be, the Emperor is genuinely upset at what has become of the Imperium and is slowly trying to put it back the way it was before the Horus Heresy. Predictably, the Inquisition has other ideas on that.
Self-Serving Memory: Upon hearing about the Astral Knights actions against a Necron World Engine, the Emperor claims he totally founded their order despite dismissing them as having a stupid name earlier.
Surrounded by Idiots: What the Emperor thinks of his immediate subordinates. This sentiment extended even towards the Primarchs, both loyalist and heretic. The sole exception to this is Malcador the Sigilite.
The Reveal: The Custodes who has been chatting with the Emperor the whole time is actually the Captain-General, leader of the entire Adeptus Custodes. He manages to use this power to dissuade Karamazov from burning the High Lords.
Magnus the Red certainly feels like he was this when he appears in episode 13.
Unwitting Pawn: Magnus is this to both Tzeentch and Horus, and is forced to accept these facts by the Emperor in episode 13.
Waxing Lyrical: Upon hearing about the heroics of the Astral Knights, he says the were "... The best around. Nothing was ever gonna keep them down."
Yes-Man: The Custodes who guards the Emperor. In the Emperor's own words, he's basically a brainwashed idiot who can only gush about everything and do as the Emperor tells him. Being the Captain-General of the Adeptus Custodes, a High Lord of Terra, politicking is a good part of his job alongside guarding the Emperor.