Don’t know where to locate the Koseki?

let me explain, I was born in japan after my birth my mother (Japanese) left japan to my father (not Japanese) country that was about +30 years ago.
My mother contact with japan was lost over the years - not sure what happened, my father passed way and my mother health is not good and it is very hard to communicate with her, I have trying to find any existing relatives to notify about her condition ( I do not have any record of them address/phone that is if they still exist...) by trying to locating her Family register - also I need my birth certification as I have none to show.

The problem is I don't know in what city hall office to ask the Family register/Koseki from ? my mother is very unclear about her last residence location, tried to trace her locations over the years - as I understand he lived with her family in Tokyo (Shibuya area) then move to Kobe city, then back to Tokyo to ota or Kanagawa area (I think she said Yokohama - but very unsure may be number of places) , so my main problem is I don't know from what city hall to ask the Family register/Koseki.

My questions:
1. Does the city keep history record of the residents? So for example if I know my mother family lived in shibuya can I go to the shibuya city hall and ask the family register that existed when they lived there ? (may be not update ...)
2. Does the Family register / Koseki contains the full family record - for example my mother, father and her sisters/brothers details?
3. Because it was many years ago (+30 years ) I don't have any address and my mother does not remember due to health issues, can I get the Family register/Koseki without the address ? Any by the family name and her name?
4. Is there central records of all Japanese citizens kept by the japan ministry that I can ask to locate the last address of my mother in japan?
tried to locate the information from the Japanese�fs web site/ other resources in the net but did not get any clear answer as I planning visit to japan I need better information where to go that will help to focus my search/area.
Right now at lost so I appraise any kind of help,
Thanks
Rony

First of all, koseki (family registration) is completely different from juuminhyou (residence registration). Typically, people keep their koseki as is, and only move it when they get married. Have you asked your local Japanese Embassy? Because that's the best place to get started.

1. Does the city keep history record of the residents?

The city hall keeps record unless the record holder requests otherwise. The record holder of a koseki is the head person of that particular koseki, which is typically the eldest person of the household.

So, yes, you can try contacting Shibuya Kuyakusho to see if they still have her record.
https://www.city.shibuya.tokyo.jp/eng/index.html

2. Does the Family register / Koseki contains the full family record

Koseki contains the full family record, but a grown family member can leave her/his original koseki to obtain her/his own koseki. But even in that case, the old koseki will have information about when and why that person left that koseki.

3. Because it was many years ago (+30 years ) I don't have any address and my mother does not remember due to health issues, can I get the Family register/Koseki without the address ? Any by the family name and her name?

It would be a tricky task, and at least you need to be able to prove that you are a close family member of your mother. It would be very helpful if she knows the address of the home she was born in. Also, if you can find her Japanese passport or any sort of official ID issued in Japan, that would be a great help.

4. Is there central records of all Japanese citizens kept by the japan ministry that I can ask to locate the last address of my mother in japan?

To be precise, there should be a record of your mother presenting her passport to leave Japan the last time she did. But I'm not sure if it's easy for anyone to obtain that sort of record.

Actually, websites like ancestry.com are helpful when trying to obtain record of when and how people immigrated to the United States. I wonder if the country you live in has something similar.

you don't have the birth certificate in US.
what document is equivalent to it in US ?