First, do you have wifi access? When you use the [name]@kindle.com address your books will be delivered with Whispernet at a fee of $.15 per megabyte or $.99 per megabyte depending where you are. If you have wifi then you can send to the email address [name]@free.kindle.com instead without any fees (both emails send to your kindle and are automatically set up by Amazon).
Second, the @kindle.com (and @free.kindle.com) email account for your Kindle is a version of whatever email you used to set up your account, [name]@gmail.com in your case I assume from your post. If you are using the [name]@free.kindle.com address to send books then you'll get an email from Amazon to your gmail.com account telling you that the book will download when you turn on wifi and including a link if you want to download with a cable.
If you've been able to set up calibre to send emails to your gmail account, then it should be easy to add the [name]@free.kindle.com or [name]@kindle.com email in the 'Sharing books by email' section of Preferences and using that address calibre should be able to send directly to your kindle.
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