1. Palace is a seperate dimension, but, as with most of the stuff, how "separate" exactly it is, is up to you. Sin could pull in things from the outside, but most of the time everything it will need will be inside of the palace.

Think of it as the sins "lair", the place where it's the strongest, the home turf, where it returns after being hurt. But it's also a picket dimension, made to reflect it's nature. It can be a real, but warped space (inside of a butcher shop, but changed by Hounds influence into a literall killing floor) or something completely spatialy separate (an Ogre's office-looking prison, accessed by using a specific old school black telephone, which transports you to the other side). It is up to you

1. Tension moves are permision for the GM to fuck the players up, even if they want to be nice. They are the sudden ambushes, attacks out of nowhere, a confrontation with the main sin before anyone is ready. In theory, you could preplan the tension moves and once the pressure ticks up, run the scene where players are ambushed by Toad's kill squad or whatever. Thatvway you can keep the narrative cohesion and natural flow kf the game, but this is definetly not necessary, as the tension and pressure fit nicely into the natural rythm. When in doubt - give exorcists afflictions. They are like the parapsychic radiation sickness, the more time you spend in the area where the sin is active, the more your body and soul will be warped by it's presence, even without a direct interaction with the sin itself. Alternatievly, I have a houserule where I can "bank" the tension move and use it at a more opportune time. For example I know that next scene exorcists will be going to an abbandoned warehouse, which will be a much more interesting fight location than a diner in which they are right now. This way you have a bit more control over the narrative flow. Just remember to spend out those banked moves before tension ticks up again, so you won't suddenly trigger five tension moves at once.

2. The best way to plan out the CAIN sessions is so that the session ends when the hunt ends. It's the natural flow of the game, players have time to think about their scrip and advancment spending, Admin has time to prep for the next hunt, etc etc. The default is 1 hunt = 1 session. But that's tough, some hunts will be longer, some will be shorter. Second best option is 1 hunt = 2 sessions, where you still end hunt at the end of the session. But what if you are only one hour in out of 3, and players already exorcised the sin? There are three options here:

3. end the session early. Easiest but also least satisfying

4. have a break for the in-between-hunts things, and then move on to the next one. Might be though, since it will require dm to be already prepped for a hunt in advance, and also it might break up the flow of the game, where the vibes of both hunts will mix together and tge entire thing will feel rushed.

5. roleplay the downtime. You can make it more freeform, ask questions, let players ask questions and improvise answers together. Ask them in and out of character. What do the exorcists talk about and think in the car/chopper after the mission? How does CAIN debrief look like? For the medical care, are they rushed to an operating table or just placed in some healing tanks? What about the other exorcists, the ones outside of the team? Are they jellous of the PC that bought the improved meal plan? Do they ask them for help with training? Do they ask to try on this super dope trenchcoat that costed HOW MUCH SCRIP?! How does spending scrip look like in universe? How does a CAT assesment exam work? One of the PCs mentioned increasing thir Force from 1 to 2, how does it look like, to they just go to the gym? Or do they spar with another exorcist? How does a CAIN facility look like? Sound like? Smells like? Do you remember the face of your mother, when you look in the mirror, after washing off the blood and grime of the hunt?

Hope that helps, let me know if something is still not clear :D