Yes you can. Goto settings/accounts & import/send mail from another address/put in your name as you want it shown and the
[email protected] which is the new email that you are going to be using.
Then go into your webhosting to create your domain email. Once you have done that go into forwarding and put the email you want to forward and where you want to forward it to which in this case would be your gmail account.
You will then verigy it in gmail. Once you verify it then make the new email your deafault so when you email out you can use your main address without having to switch back and forth between the paid and gmail. This is also done in the accounts & import