NO.
Yes, you can.
A passport.
Your question does not make sense. You need to be a US citizen to have an American passport. If you have a green card and an expired Russian passport, contact a Russian embassy in the US and they should be able to issue you a travel letter or stamp your passport so you may travel back to Russia. But returning to the US is another problem.
You must always use a valid American passport when entering and departing the US even if you are a dual citizen.
I live in the USA with a Green Card but am a citizen of the UK and a UK Passport holder. I am thus to the USA/US Immigration a foreign passport holder.
Passports aren't issued by cities, they are issued by countries. In order to have a passport, you need to be a citizen of the country you are applying to. So if you are a British citizen, you get a UK passport, if you are a Canadian citizen, you get a Canadian passport. It doesn't matter where you were born, it's what country you are a citizen. If you don't have citizenship then you cannot have a passport. Ex: A Canadian citizen, who is NOT an American citizen cannot have a US passport.
If your passport has been stolen, you won't be able to travel.
Whether a US citizen can work in Japan is completely unrelated to that person's UK residency status. It is a matter for Japanese law and regulations.
Yes. As long as the passport is still valid.
Yes you can, as long as your passport is valid
Yes, with a passport you can fly from the US to anywhere.