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Yes, but you need to jailbreak your Kindle to install it. If your Kindle is recent or anywhere near the latest release - it can't be rooted.

I think you can do this with a stock Kindle, by sending the epub to the email address of your Kindle (which you can find in your account settings).

No, epubs - presumably as a competitor format - are not supported.

You can do it (email "conversion" - often just a horrible mish-mash of text and images) with PDFs and text etc.

Epubs need to be converted first by a third party service, eg calibre.

Where by “competitor” read “defined and standard”.

Yup, it will.