Welcome to GTD! I'm sure all of us here appreciate how much smoother your house sale will go with a system like this. Well done. Now,
> When I am processing my inbox and I come across a project - Buy New House - which I am currently doing, I put it on my Personal Projects List as Buy New House.
Good - It'll take more than one step to accomplish, so it's a project. Put an entry on your master projects list (personal, if you've split personal and professional), and make a folder for it if you have some paper to file. Use the same name as the master entry. In this case, you know you'll have plenty of paper, so go ahead and create the folder right away, if you'd like.
> When I do this I naturally start thinking of a million things I need to do. When I make a brainstormed list of everything I need to do where do I put this and how do I process it?
Put it in the folder! If you use an electronic tool, put the file in a folder on your disk with the same name as your project. As ideas come to you during the day, or when (say) meeting with your partner, put them in the folder as well. I keep it simple and use a legal pad for notes.
I wouldn't break the project into separate parts at first, unless you decide you want multiple big pieces moving ahead simultaneously. But you will want to do some planning of steps to take, e.g.,
o research realtors
o select realtor
o meet with her to decide when and how much to list for
o list house
o wait for inquiries and show house
There are phases when you'll be active, and times when you'll be waiting. And you'll want to use your Waiting For list and calendar, of course. Waiting for inspector's report, waiting for buyer's response, open house date, closing date(!), etc.
Finally, you'll probably have an overall plan for the project, commonly kept at the front of the folder. It'll have major actions, milestones, status, etc.
Hope that helps; good luck!