You cannot email an AZW3 to your Kindle, although you can sideload it via USB. I assume you mentioned send-to-Kindle because you want it to whispersync between devices?
For a partial summary of the issues with AZW3/MOBI, whispersync, etc. see here:
https://www.mobileread.com/forums/sho...17#post3042617
Depending on the formatting, EPUBs may simply have been an automated conversion from DOCX, which may or may not be good quality. Assuming they worked on the EPUB version, it will probably be a better conversion, yes.
Putting stuff in folders -- I assume you mean collections? There is a Kindle Collections plugin for calibre, which requires being jailbroken.