(a) as a US person working in Japan for a private or local entity, you should be eligible for Foreign Earned Income Exclusion. For this you need to enter your income ( gross / before taxes ) under Personal Income tab and Foreign Earned In come Exclusion. This will walk you through filling out form 2555. See details of eligibility here --> Foreign earned income exclusion | Internal Revenue Service
(b) any un-excluded income would be taxed by both US and Japan and therefore is eligible for Foreign Tax Credit under the Tax Treaty between US and Japan.
(c) Here is some information on Social Security --- you need to pay this to one country ( suggest choose based on your long-term goals i.e. where you are going to retire ) --- Totalization Agreement with Japan | International Programs | SSA
If you need more help on this , please add to this with questions/ comments and I will circle back.
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