If you are over 18, you should be able to apply for a new Japanese passport without having to tell anyone. You will need some form of birth certificate, so I suppose if you are in the US, the Japanese embassy will want to see a copy of your koseki. I don't think you will be able to retrieve a copy of your koseki remotely. You would need to check with the city/ward office of your last domicile to see what they require. Nowadays the ward offices of Tokyo allow to get copies of your koseki through a machine (sort of like an ATM machine, but one that prints out official documents). I think most major cities/wards have this system. If your last domicile was a remote or rural location, it probably has a much more analogue system of recordkeeping. Anyway, if you are over 18 you need not worry about your adoptive parents finding out - its the bureaucracy you have to worry about.
I haven't heard about the situation where you describe (adopted kids having an independent koseki set up for them). I'll dig around a bit, but I couldn't find anything when doing a quick search.