Questions about the Kindle Send-to-Device service?

The DX is special–as it doesn’t have wifi (only 3G), you’ll have to pay for any transfers via Wispersync. On all modern Kindles wifi can be used which is free.
As you are already converting from ePub, the simplest solution is to use Calibre. Calibre does an excellent job of converting and makes it easy to send the books to your @kindle.com address for cloud syncing.

Your Kindle has an email address you can use to send stuff to it. (You will probably need to whitelist the sender first; go to your account on Amazon to set this all up.) It will also end up in the personal document section of your Kindle library.
It’s free for wifi but will cost money for 3G… $0.15/MB in the US, $0.99/MB outside of it is apparently the current rate for the DX.
Also, the DX might be too old for whispersync of personal documents anyway, in which case it will just be a one-time sending process, and you might as well just use USB instead.