Ah, from your post I understood you wanted Calibre to check your mail. As I understand it now, you can send them to your Kindle in 2 ways; one way you have to pay and the other one is that you send them to your gmail account and download them from there.
I'm not much into the Kindle but this seems a bit strange to me as it would be very easy to avoid data charges. I can imagine they blocked downloading books from your webmail. Perhaps this is a stupid question, but have you tried emailing a book to yourself and download it from your webmail. If that works, it'd be worth investigating automation.
Also, can't you just put them on there with a usb cable? Seems a lot easier to me
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