Even this is just my personal opinion , I had used Free hosting last year for my Blog , Suddenly I got massive traffic on my Blog . But hosting provider unable to bear by Traffic , So in my suggestion if you are looking for earning money with our domains .... forget the free hosting
You can register a domain with web host as you sign up, you do get a free one with your hosting for as long as your hosting is with service provider...
Can somebody tell me a free domain name host with 1 GB disc space, php, imap, and no ads?
I don't recommend to get a free domain with a hosting. Your Domain name is the most important asset of your online business. Get a domain from Godaddy or any other domain provider and look for hosting separately.