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1. The Product Red iPhone 7 was a limited edition iPhone released in March. Apple stopped selling them in September.
As its manufacturing date precedes that of iOS 11, and even iOS 10.3.3, its not surprising it has iOS 10.2 still if its been sitting in a store or warehouse for several months.
2. All Apple iPhones carry the label "Designed in California, Assembled in China". All iPhones and most Apple Products are assembled in China.
3. The No Sim message is pretty standard when there is no sim card installed in the iPhone.
None of those things even begin to hint at it being a fake product. In fact they all pretty much add to the not-fakeness of it.
4. The battery issue is a little confusing, can you maybe clarify it?
Other than that, you can go to Check Your Service and Support Coverage - Apple Support and enter the serial number found in Settings->General->About->Serial Number to verify its authenticity.
Question: My IPhone 7+ Product Red