If i understand you correctly, you inherited accounts from your relative that were closed due to a death claim and then you reopened these in your name. In this context, you will report these accounts only once since these were the same accounts before and after the death claim.
Yes, since you were the heir of the accounts prior to a death claim being submitted, i would report these accounts as your accounts. it would depend on that countries or how the countries would treat the ownership of those accounts. If you were declared the owner before the death claim, then yes, you will report these accounts on your 8938 as your accounts as well as an FBAR/FINCEN.
As far as an 3520 for the gift tax, generally inheritances are not considered gifts unless you were in receipt of the gift prior to the passing of the deceased relative. A POA does not qualify as being in receipt of the gift. Please read this IRS FAQ for further verification. You will file an FBAR/FINCEN on the accounts you have inherited.
Lastly you mention you will file a 3520 and FBAR/FINCEN due to my account balance and the POA while she was still alive. You will not file a 3520 or an FBAR/FINCEN on accounts that you are a Power of Attorney because you do not own these accounts. You are simply a custodian not an owner for your non-resident relative, so in no case scenario would you file a 3520. You would file a 8938 and FBAR/FINCEN once you become sole owner of those accounts.
Keep in mind also you will file a 8938 if the aggregate amount of these inherited accounts is over $50,000. The FBAR report must be filed if the aggregate amount is over $10,000.
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