Mandrill automatically adds any domain names that are used for outgoing mail to the Domain list that you are seeing. This is normal.
I should make it clear that Mandrill is NOT using any Gmail server to send you message. The message is being delivered through the Mandrill system alone.
Once you get your custom domain you should begin using that as the "from" address for your emails. This domain will then show up in the domain list.
I highly recommend that you setup the proper DKIM and SPF records in your custom domain DNS records. Mandrill has very clear instructions on how to do this. Just click on the "View DKIM settings" and "View error details" in the domain list.
I recommend you check out the following page for additional information: http://help.mandrill.com/entries/21681347-How-do-I-set-up-sending-domains-