Kindle Fire has a number of issues related to sideloaded books. I may be wrong (I don't have one and all knowledge comes from discussion here on MR) but I don't think the Fire can read covers from sideloaded books.
Also, if you are downloading via the browser, you should be able to download AZW3 as well. You may have to save the file to storage in the documents folder, but I would think that the AZW3 handler is declared by the reader app?
If you are using Send-to-Kindle, covers should be sent from Amazon's server together with the book. (I am not sure what you mean by non-Amazon email.)