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Hayato took the bullies' side and caused Yukino's individuality
It's been hinted several times that Hayato made a mistake in the past which cost him the friendship he had with Yukino. As Yukino explained it herself, the cause of it was jealousy. She was too perfect for the other girls and these tried to made things fair by making her life harder every day. But Hayato never helped her and decided to abandon her instead since taking her side would have ruined his social image which is the thing he cares for the most, even more than Yukino. This forced Yukino into never rely on someone again as her trust with Hayato have been destroyed and took everything on her own hands while growing to hate Hayato for chosing superficial relationships over her.
Hayato is/was Yukino's crush
Possibly way off the water, and possibly reached Pluto on this WMG. But the series does seem to imply this.1a) The Hayama family and the Yukinoshita family are very close to one another, and Hayato and Yukino are both in the same grade. Perhaps they wanted an arranged marriage between the two?1b) Alternatively, the families wanted Hayato to marry Haruno, as Haruno is the eldest child of the Yukinoshita family and is the more preferred child, causing Yukino to yearn to escape her sister's shadow by being the best at everything.2) Hayato is jealous of Hikigaya for being able to achieve his goals, as Hayato does not want to seem to prefer one side than another.3) Yukinoshita aims to improve everyone, rather than to let everyone undermine her. This causes her to be at ends with people like Hikigaya and Hayato, who tend to remain static after solving problems.4) Combined with WMG stated above - At one point, Hayato showed his true side to Yukino, but betrayed her when he took the bully's side. Why the two of them still know each other is through family relations.
Therefore, I conclude with this. Yukinoshita tends to grow more warmer with Hachiman not because of who he is, but because of who he reminds her of. Hikigaya has not a mask to hide behind unlike Hayato. However, Yukino does not want to reveal her feelings to him in case she gets hurt again, but will reveal her feelings in the end of the series, as she knows that Hikigaya will not betray her.
He never does fully explain why he was ridiculed by his classmates through all of his elementary and middle-school years. His only excuse is that he is seen as 'gross' by others, but there is nothing to back-upon his claim.
Jossed. His interactions with Kaori during the double date are evidence of his claims.
Origins of Hikigaya's nickname Froggy-kun
Throughout elementary and middle school, Hikigaya has been ostracized by his classmates for simply being 'weird'. Although not much of Hikigaya's life during elementary and middle school is revealed to us, the audience, we can deduce or WMG his situation with the facts given.
1) His parents are not always home.2) He has his younger sister to take care of.3) His classmates created an imaginary virus after him.4) The same classmates also gave him the nick-name Frog.5) Girls generally find Hikigaya to be disgusting.
I believe that Hikigaya's nickname Froggy-kun comes from an insult made by the rest of his classmates due to his constant absence from a disease from when he was younger. Perhaps in his earlier years, this sickness made him sound like a frog and he had to stay at home for long durations of time in order to recover - only going to school sparingly. However, since his parents were gone frequently, Hikigaya had to miss out on school for long durations of times in order to care for his younger sister.
Since Hikigaya did not come to class frequently, the kids started making rumors that Hikigaya was part 'frog' and that he had to leave school to find a cure to become human. Since kids are idiots, they also thought that Hikigaya's frog disease was contagious and nobody wanted to stay with the boy who transformed into part-frog and wanted to be a frogman like Hikigaya.
The girl who will eventually win Hikigaya's heart is a Totsuka.
Not his old friend Saika but Saika's identical twin sister who is just as sweet as he is. The only thing holding Hikigaya back is the fact that Totsuka is a boy. What if there is an equally attractive Totsuka who is a girl? She would sink his battleship in one shot.
Hachiman will become the new student council president in his 3rd year
The opponent he'll be running against is Yukino, who falsely assumes Hachiman won't make a good president.It ends with Hachiman winning votes from people who he helped over and their friends.Yui will be more torn during this arc, feeling she fails as a friend.