We normally see him at his angriest, or at least the most emotional, whenever he's being reminded of his past in the port mafia and more specifically what he had done. For example in the guild arc his dislike for Yumeno after coming face-to-face with them for the first time in the years was very apparent and filled with hostility as they're probably a representative of Dazai's past mistakes (he was earlier reminded about his past being mafia black according to his records). So when Jouno points out his rage when he's being arrested due to his past crimes in the port mafia it's likely him being angered from how the worst parts of his past always seems to haunt him. Dazai is not really the type who likes to confront his past, which is understandable, and so his hidden anger stems from those reminders that makes him remember what he's been trying to move on from.

I think he does, he never shows discomfort when Nikolai held his hands or when he's talking to him and he doesn't seem to actively try and distance himself from Nikolai. He also doesn't correct Dazai or seem bothered when Dazai says "you have a great friend" when Nikolai proposes the prison escape game, he even agrees with Dazai. There is also the fyolai omake though I'm not sure I'd consider that as canon, or has as much weight as the main manga. I would describe Fyodor as having a certain fondness or interest in Nikolai but currently not much more than that from what we can see.

There are likely multiple reasons why he doesn't openly reciprocate any of Nikolai's affection, such as his trust issues over the things he can't control (I think it's been pretty well established Fyodor can't control Nikolai in any way), and the fact he probably doesn't see the need for "friends" since he might not see having "friends" in his near future due to his goal that he wants to achieve.

I think the confusion comes from the fact we've seen moments where Nikolai visibly hasn't had his cloak on him and him still using his ability, for example when he's disguised. Those moments could be explained by the fact he probably had the cloak hidden under his clothes. Though that then doesn't explain how he did that trick to fake his death during the sunday tragedy where he didn't have his cloak on him. I think how his ability works is that he doesn't necessarily have to be wearing it, just that his cloak is a /literal/ portal (basically his cloak can act as either the exit or entrance for the connection he wants and is also the setpoint for that 30 meter distance he's restricted too, meaning that his ability is to activate his cloak's spatial connection to a pocket dimension so that he can use his cloak to set spatial connections within 30 meters of where the cloak is without having it on him. This is then why he was able to do that trick in the sunday tragedy, get fyozai out of their cells and to hold himself up with that bar by using his ability by taking off his cloak and placing it on the ground (the cloak opened a connection between the area he placed it down in and the area where he's currently holding himself up).

It is exactly as how he said it in that one chapter when we're first explained his ability, he does need his cloak because it's what's connected to that pocket dimension, but he doesn't necessarily have to be wearing it since he never specifies that.

I was going to bring up Fyodor's weird wealth of knowledge he seems to have in your previous question if I hadn't seen this one. He might know something about abilities that we don't given he seems to weirdly know a lot about Atsushi and his potential connection with the book, and his ability also seems to have weird properties compared to most other abilities (either because his ability has something unique to it that we haven't seen before (which is mostly what I suspect) or that his knowledge of abilities allows him to have that odd connection with his). It's kind of unsurprising that he seems to know so much since his end goal is connected to abilities (and the book) in the first place, so I can imagine maybe after he manifested his own ability he started to learn every thing about them, for many reasons we could speculate. Fyodor remains an enigma.

If we're thinking about singularities in bsd metaphorically, if abilities in bsd are part of and reflective of the user's soul then I imagine singularities as sort of a manifestation of the user's most intense feelings, bringing out the most intense part of themselves, which then makes them feel less human, or I guess more disconnected from other people, because that ability part of themselves is amplified if that makes sense? (because there is already a separation depicted in the series between those with abilities and those without). They basically become more ability than human, which is what Verlaine and Chuuya were treated as, Shibusawa kind of lost himself after Fyodor activated his singularity, and for Dazai it brought great sorrow because he knew he didn't really have a place in his world.

It took me a while before I actually started shipping fyolai because at first I didn't care much for Fyodor's character. It kind of took me joining the fandom, and then me taking more interest in Fyodor, for me to start paying attention to their dynamic and then start shipping them. So I guess my first impression of them wasn't really anything remarkable, it was clear to me they both liked each other in their own weird way but at the time I didn't have much interest for them back then. (and I'm not someone who's very crazy into shipping as well so I already didn't have much interest in trying to ship the characters when I first got into bsd except for maybe skk)

Don't worry you're not the only one who's noticed! Yes after this chapter and then rereading chapter 101 there is certainly something that feels strange about it. You can't attach images on here so I'll post screenshots on the tweet but yeah it does seem like genuine shock coming from Fyodor. He even panics when he sees that the door's closed and gets Chuuya to try and punch the walls, makes you question why would he be visibly panicked if he supposedly prepared for something like that to happen? I do talk a little about this here as well: https://twitter.com/VannyStarBSD/status/1621547067792400386?s=20&t=t63CGvqFtU_KCBwVjGIFSQ

It is possible that maybe the reason why we don't see the door deformed is that there might've been a continuity error, but his initial panic does seem genuine

If he was lying though, then I'm not sure how else he would've gotten out