Well, his friends might be sympathetic to him. They could try to comfort him and maybe even stand up against the teachers on his behalf. This would strengthen their friendship.
Some other students might take advantage of his situation. They could spread rumors or isolate him further, thinking that his hatred for teachers makes him a troublemaker. This would create more conflict in the story.
One possible reason could be that he has had bad experiences with teachers who underestimated him. For example, they might have not recognized his potential and discouraged him from pursuing his dreams, which could lead to his hatred.
In the fanfiction, Izuku may start to undermine All Might's actions. He could try to expose any flaws or wrongdoings of All Might. Their relationship goes from one of idolization and guidance to one of opposition. Izuku no longer looks up to All Might and may even try to bring him down. For example, he could spread rumors or try to find evidence against All Might if the story goes in that direction.
At first, it could be full of hostility. Jirou would see Izuku as a threat to everything she stands for as a hero - in - training. Izuku would see Jirou as just another obstacle in his path. But as time goes on, they might start to understand each other's motives better. For example, if Izuku has a legitimate grievance against the hero society that led him to become a villain, Jirou might start to question her own beliefs about heroes and villains.
Their relationship could develop slowly. At first, Toga might be curious about Izuku because he's so different from the others she's known. Izuku would be cautious but also see that Toga is more than just her villainous reputation. They might start by having small conversations, gradually learning about each other's hopes and fears. As they spend more time together, they could develop a deep bond. Izuku might inspire Toga to be a better person, and Toga could give Izuku the confidence he needs in a world that doesn't believe in the quirkless.
Another aspect could be the general discrimination in the BNHA society. The fact that society is so focused on quirks and those without are often looked down upon might have made Izuku bitter. He could be hating everyone because he sees the society as a whole being corrupt and unkind, not just the individuals who directly wronged him.
Well, initially, Izuku might be overwhelmed by his new cosmic quirk. But as he trains, he becomes more confident. For example, he could start by accidentally causing small cosmic disruptions and then learn to control them precisely.
In certain fanfictions, the izuku system is a secret. This secrecy can strain his relationships as he has to constantly lie or be evasive about his sudden improvements. But when the truth comes out, it can either lead to a huge rift or a deeper understanding among his friends, depending on how the writer chooses to portray it.
It might make him more hesitant to form close relationships with adult characters. He may be guarded and not open up easily to them.
All Might would be furious. He would see it as his failure to protect Izuku. He would probably mobilize all the resources he has, both in his hero form and his connections in the hero world, to rescue Izuku. And he would be blaming himself all the while for not being there when Izuku was kidnapped.
In many adopted Izuku fanfictions, his character develops confidence. For example, if he's adopted into a loving family, he no longer has the self - doubt that plagued him before. He starts to believe in his abilities more. This new family support gives him the boost he needs to train harder and become more assertive in his hero journey.