The free domain will be registered on a year-to-year basis. How fair would it be if you registered for five free years ahead of time and then left dreamhost after 6 months?
At the end of each year, you will have to go into your control panel and renew it, but you won't actually be charged for it, since it's included.
If you've received confirmation from DH about your registration, you should be okay... sometime the system just has some lag.
As mentioned, if you want official word from DH you need to submit a request to support through your web panel. Be sure to specify in the "category" field that it's a domain registration question.