I think him and Killer were child soldiers. Killer even has the title massacre soldier. And Kid has CoC. So far almost all CoC users are people who were shackled/enslaved/oppressed by sth or somebody in the past and wanted to break free. It would be fitting if they had to fight for a country and people who discriminated them and that‘s why they hate everything to do with authority and view innocent civilians as enemies, too.
I agree.
I think Killer is named like that cause when Kidd met him in his childhood (as both child soldiers) Killer was an abandoned child, like most of the other kids, and so he didn't have a given name. Seen that he was the most lethal of the kids, he was named like that by them.
With Kidd's arrival his life took a different turn and was probably inspired by him, as the first one who really treated him as a friend. Kidd possibly teached him how to be proud of that name.
To make it brief, I think their backstory is about child soldiers used by adults to fight wars, but with Kidd showing up as a rookie he rebelled against the adults and the other kids stuck with him. That's probably how he got the alias
'Captain'.
Just my 50 cents but that's my headcanon for now, based on a lot of hints.