You asked so my brain has to answer.
Sirius is still super pissed at his family and the ways of the purebloods and told everything about it to Severus, so Severus is not particularly dreaming of being part of such a corrupted elite. It took time for Severus to open up about his home life and real living conditions in the Muggle world (since Sirius was so prejudiced and ignorant and was ashamed of it). He did tho during their third year and also educated Sirius about it a bit more.
Both of them are pissed at the world. Isn't there any good place anywhere? Big rebels energy.
Sirius's relationship with Regulus is better because not only did he spent more time with Slytherins to hang out with Severus but he also befriended a few of them because charisma and aristocrat cred despite being a gryffindor blood traitor. He's alarmed at Voldys ideas spreading around and him and Severus are talking about it a lot. Both of them find it kinda sus from the beginning since they're both disillusioned. Severus seems a bit tempted sonetimes but Sirius drowns him again each time with stories about the things purebloods are really saying about half-bloods and muggleborns and is adamant he's never going to be accepted by these specific people (which upsets Severus because where the fuck is he gonna be accepted then? 'By me always' answers Sirius each time with fire in his eyes)(Severus has to tear his eyes off after a few seconds because he's so intense). They argue with Regulus about Riddle (Severus is sitting between them in the Slytherin common room and they're standing, shouting at each other, this is a normal occurrence but not alarming by any means). Regulus is still quite motivated to join because he wants to believe Riddle is good for the wizarding's society. (He will join but regret it faster than in canon.)
Sirius is still super tight with James. For some periods of time Severus and James are equal in Sirius's heart before it slowly shifts with puberty and hormonessss (also just Severus being super duper smart and having a wicked humour Sirius absolutely adores). James is the kind of pureblood Sirius really appreciates because he's not stuck up with tradition, tho he's still kind of an entitled twat - which Severus points out and Sirius brushes off for a good while. James and Severus don't get along super well but it's... manageable with Sirius in the middle. Tho it happens quite often during exam season that Severus will be really withdrawn because he's studying like crazy by himself (of course Sirius and James are brilliant but also super careless about passing because well, why should they care? They don't even need to find a job at the end of the day). This will not only push them apart and create a few fallouts when Severus gets too annoyed at Sirius and calls him a fucking cunt and reminds him that he (Severus) doesn't have a choice. Sirius will come around because he's a hard-headed bastard but he's capable of self-reflection way more since he's not just enabled by James all the time.
Basically by being close to Severus, Sirius will be less of a performative rebel and really realize a bit more the true hardships and sometimes misery of non-purebloods wizards.
About Remus, Severus would be told about his conditions pretty fast and would be super alarmed because what do you mean a werewolf??? After the initial shock I think he could get along with Remus because they're both a bit more withdrawn and are both -for different reasons- on a more struggling level in terms of future. They're a bit more level-headed. The difference is that Severus never misses an opportunity to remind James that he's acting like a privileged twat whereas Remus won't dare for a long time (and Severus thinks he's acting like a cowardly wet mop). He will get bolder to his contact tho. Severus faces his future with an iron determination (hiding quite some doubts and fears) and he's also not shy to just address the fact that Remus will struggle too and he has to prepare himself for it. They end up studying together and start coming up with strategies to get around the system somehow.
Sirius would get maybe a bit closer to Remus because of that as well. He'd have to face Remus's status and difficulties more upfront because of Severus.
Really, Severus is the one always reminding them (and they're sighing) that the world is shit and the future can be bleak. With time Sirius will understand more as Severus tells him about his family. And with time he'd have to admit he's not all wrong especially with the war looming. Severus is telling him about what's being said in Slytherin and who is thinking about joining.
When the boys start studying to become Animagus to be with Remus, Severus is adamant that this is extremely stupid and reckless and he won't take part in that. The 'being with Remus' part only tho. He will also try the Animagus thing because Severus is never backing down from a magical challenge. Also he likes to be there to keep an eye on them (James is pissed).
If he manages to do it he'd either be a black cat or a raven. I'm leaning towards raven, since he's shit on a broom he'd really enjoy flying like this. The first time he'd land and transform back, he'd look so thrilled and flushed with pleasure and alive that Sirius will kind of fall in love on the spot.
About Peter. I think Severus wouldn't like him. He'd be cordial with him but would absolutely despise the way he's gaping at James and Sirius (also maybe a tad jealous at some point tho he can't figure out why)(Peter looks almost like he's crushing on both of them). He knows Peter is a follower and that's fine but his guts tell him to not trust him fully. They'll never be close. Same for Sirius tho he'll trust Peter way more and love the attention.
Lily and Severus remain close-ish tho they're still drifting apart with time. Severus clings to her quite a lot but gets more and more frustrated as she's changing. Sirius is kinda pissed at her (he has nothing against her per say but he's not fond of her either and the fact that both Severus and James gives her so much attention where she's not treating any of them really well and it's pissing him off). So he's the first one to complain out loud about her when Severus is sulking and it actually... kinda helps Severus finally (with time) opens up and complains as well (not at first because he'd defend her). Sirius helps him come to terms with the fact that their friendship is dwindling and it's not entirely his fault. Severus will struggle to accept it and get extremely silent when she's around.
With time he'd simply ignore her. If Severus and Sirius start being closer with each other (read: gay) during their last year I can imagine her somehow coming back to talk to him and they maybe talk things out (read: big fight)(Sirius is super proud). They manage to find ways to be cordial since she's getting with James and Sirius is still quite tight with him.
Now where did the bullying go? James is still a fucking twat and very anti-slytherin. He'll find people to hurt. Sirius, since he grew up in a violent environment will find it funny. Severus not at all. He'll absolutely talk about it to Sirius and this also will cause quite a few rifts. At the end of the day the House rivalry and the political climate can't be avoided. Sirius will participate in it for a good while. He'll almost lose Severus's friendship over it during their 5th year. Big drama. Because 'How are they different from me!?' 'You're not like them tho!' 'Aren't I? Just because I don't think about joining? We both fucking know Potter hexed him because he's lower-class! He never hexed Goyle or Evan!' 'You're talking shit!' 'You're so fucking infuriating Black I can't stand this act of yours anymore!'
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