Shino had to change a school, because being bullied- it was before the incident with Endou.
I think, it would summarize, if we answer with "a big part of Earth population are *hit", but, generally, if Sinon says "would you hold a hand of a murderer"- it seems, that even if "you" defend your life in Japan, you are still seen as a criminal, if you harm the offender.
And killing somebody was quite a strong matter that bothered people around Shino. It seems to be as:"But she kills people- I am not sure it goes fine with me".
As for the second incident: in the book, it is said that Endou pretended to be Shino's friend, because she knew that Shino lived alone. Then Shino once came home and realised that Endou and her followers invited some males to Shino's home- it caused her shock, so Shino called the police and insisted she didn't know anyone from the people, who invaded her home.
The invaders had troubles, and after that started to bully at her.
In the book Shino also said that she would go to the police, if the blackmail didn't stop. I haven't read further, but it doesn't really seem like Asada really was going to do that.
Yes, it was said that Shino's childhood problems were related to her killing that bank robber. Now, seeing the young Shino as one of those psychopath killer lolis from anime might not have been the other people's fault (although her psychologists really ought to know better), but bullying somebody like that is 100% lack of self-preservation and should be punishable.
Well I think you could assume a deciding factor for the bulling in her childhood is simply that children can be really cruel and can be some real assholes. It's nothing rare that they gang up on someone because they are different, and no matter how you see it a child that has killed someone and is suffering from PTSD is different.
It's Japan's gun culture. In america for instance a kid that killed a man for thwarting a robbery would be treated as a hero. In Japan the fact she shot someone and killed them is far more of a negative than her stopping the robbery.