So I need to get the Domain name, but of course Home Premium doesn't have it, is there anyway possible like a registry edit? If not, could you please explain why it won't work? I really don't want to spend ~$100 just to upgrade...
All of the official info on joining a domain as a workstation is that it isn't possible in the home versions of Windows 7.
You might be able to find third party software to work-around it, but I don't think a mere registry hack will solve your problem.
I don't have a full on explanation for why this won't work because I barely understand and just took for granted since Windows NT 4, that certain functions just aren't integrated into the consumer distributions of Windows. I didn't pay attention about this particular one, but I knew, for instance, that you needed Windows Server or Professional (or Enterprise, or later, Ultimate) to do remote desktop access without third-party software. I don't think that those things can just be "unlocked" from home versions, and upgrading Windows or third-party software are the only solutions there.