Yes - when you select a mobi book in Calibre and select to view it then it IS internally converted to ePub from mobi before opening the result in the Calibre viewer.
As to whether the book in question will work correctly on a real Kindle I cannot say. If you want to check then you can check by viewing it in the Kindle previewer which should accurately emulate a real Kindle. Alternatively you can use the Kindle reader app for your PC/Mac which while not being as accurate at emulating real Kindle hardware should still give you a good idea as it understands the mobi format without doing a conversion.
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