White box replacements are a mix of new and refurb. If you get a refurb it is indistinguishable from new.
Apple refurbs have the casing replaced and all components rigourously tested. Anything that fails testing is replaced. New phones do not even go through this process (although they have their own Q&A process).
In addition you will probably be able to reap the benefit of a phone made later in the manufacturing process. By this point Apple SHOULD have all the kinks and issues worked out.
If you are to sell it, I would indicate that it's a replacement phone. For buyers like me that's a bonus because I know it's relatively new, but of course your mileage may vary.
On refurbs, is the battery also new as well as the casing?
And are the parts that are not new the same spec as the ones in a new phone? For example, would they use a lower quality DAC or NAND flash in a refurb?
They would not use a specific DAC or NAND in a refurb. You would get whatever happened to be on the circuit board in the refurb. You seem so paranoid about this, if I were you I would try to get them to repair what you have.
If I were buying your phone I would prefer to have the serial number mayptch on the box but it would not be mandatory.