Why are so many members of the Knights of the True Cross, specifically those outside of Japan, Catholic? The answer has to do with how centralized the Roman Catholic Church is. All across the world, all Catholic churches answer to the Pope and the Vatican. Therefore, if the Vatican wished to join the Knights of the True Cross (and it is heavily implied they are the ones who founded the order), at least all ~500,000 ordained priests in ~3,000 dioceses, if not all ~1.2 billion confimed Catholics in the world are eligible to join the Order. Contrast this to a religion like Buddhism, in which there are several different sects and leaders. The Myodha, as a small sect with only one temple, joined by themselves and there's maybe only 200 of them at the most. Shinto is even smaller: there is no overall power structure and each temple functions on its own.
Adding to this is the Catholic Church's demonology and exorcism traditions, which date back to the formation of Christianity. Shinto has a similar ancient tradition based on legends of yokai, so it's likely more than one temple is a member of the Order. Buddhism, however, does not acknowledge demonic beings, and it's likely the Myodha sect, like other Japanese temples, only picked up on exorcism due to the unique syncretic fusion between Shinto and Buddhism Japan demonstrates. It's likely Japanese Buddhists are the only Buddhists in the Order.
This also serves to explain why no Protestant members of the Order have ever been shown. Protestant Christian denominations do not accept the idea of demons (apart from Satan) or exorcism (some don't even believe in Hell). It's likely some more left-wing Protestant denominations like the Anglicans and the Methodists know about the Order and choose not to participate, but it's likely that fundamental and charismatic Protestant churches like the Southern Baptists and the Pentecostals do not even know about the Order and demons due to the threat they pose to the order; these denominations might even lynch Order members for spreading heresy if they were ever informed of Assiah, Gehenna, and demons.
Most of the drama comes from a lot of situations (especially in the manga) that could be potentially handled by just talking things out. Especially in regards to Yukio, who seems happy to Wangst his ass off instead of taking Rin's offer of help. But here's the brilliance: every single one of them are teenagers. And Wangst and poor communication is common during those years of development. Kato has an entire cast who are ultimately normal teens thrown into dangerous and fantastical situations...who react naturally according to their character and age.
Shima Renzo being revealed as actually a Double Agent makes sense if you look at a previous chapter (ch39/40) where Mephisto claims to know everything that goes on in the Academy. Combine that with his status as The Chessmaster and his general hints of being something of a control freak, and no wonder The Mole turns out to be a Double Agent. No way was Mephisto going to let them not be in his control one way or another.
More recently, Lightning makes a comment along the lines of "you can't help someone if they don't want to be helped". This is the crux of Yukio's FaceHeel Turn; he refuses to be helped, refuses to accept the truth outside of his twisted views, and thus he falls instead of finding a way to keep afloat.
Way back in the Exwire Exam arc, it's mentioned that demons go straight for Arias once they start chanting, necessitating them to either have guards or pick up fighting skills they can do at the same time as chanting. Considering they are chanting what could instantly kill the demon, this makes sense. This is a actually important come the origins of Rin as a baby. Once he was born, he just cried...until an Aria started chanting. At which point the entire army brought to kill the Son of Satan started finding out that this baby is a half demon for a damn good reason. However, as horrifying as it is for baby Rin to go Killer Rabbit on a bunch of humans just six seconds after birth, the fact he only attacks when an Aria starts chanting - and goes for him in particular - makes it clear it's actually in self defence.
Why does the Kunstliches Leben Lab have a German name when it is situated in Poland? It was founded by a disciple of a disciple of Faust, most likely by a German.
Baby Rin is actually fairly accurate. As a newborn he's wrinkly and alien looking like an actual newborn. It's normal for babies to be agitated at birth, it being a stressful situation even in the best of circumstances, and it sucks for the Order that he had powers from the get go. Even baby Rin's mobility isn't as much of a stretch as one might think, because babies are born with a stepping reflex and Rin already has super strength.
The three colors of demonic flames make a certain amount of sense. Red for the flames of Assiah it's because even if actual fire is rarely red, the color red is frequently associated with it, and demons are heavily influenced by human ideas. Both black and blue are colors associated with eternity/infinity, which fits with the Gehenna association.
The demon Astaroth had possessed a human (Reiji Shiratori) to get Rin to awaken his powers. The same demon, still in this possessed body, had been hit by a truck coming at him in full speed but had healed with his demonic powers to continue his attack. Later, the human boy is successfully cleansed and separated from this demon. So, does the boy's body stay healed? Or do those effects wear off once the demon is no longer possessing him? He is not shown to be awake after the battle. Plus, he is never shown again even around the campus of True Cross Academy where he said he will be attending soon along with Yukio.
At one point Hachirou takes control of Shura and sends her after Yukio to get herself pregnant against both of their wills. Rin saves them by interrupting, but how many women in the past had the exact same thing done to them successfully?
All of the zombies in the Inari Arc were tourists who had no idea what they were walking into and still wanted to go home.
Section 13 targeted the bottom scorers in the exorcist classes for recruitment as caretakers for clones, preying on their desperation by promising work and opportunities.
Cloning in the past required surrogate mothers to carry the baby clones to term. It's heavily implied most, if not all, of these women died in the process as Jenny mentions another caretaker who took the job and was never seen again.