@Stephen_Graham Stephen, I think your main questions are
1. Do you mind all subcontractors seeing the whole Project. If you don't you can add whoever you want to a Project as a guest member and divide your job phases with Sections. They do not have to be Team members, just read the Asana Guide on how you set up options on Public, Private, Hidden etc in Projects, Teams etc so somebody does not see something you do not wish them to see.
2. If you do mind then you mayl need to use Teams for each job and have more projects under the Team characteristic to the job disciplines such as Architecture, Electrical, Plumbing etc. You could stick with just projects and limit outsiders to the task only within the Project but they will not see the name of the Project and it will only show up in their My Tasks. I personally would not like that but you may.
3. Their are options that involve Project Members, Assignee and Followers but the observations and questions I posed are much the same.
Here are the critical guide extracts:
There is critical assistance in the Asana Guides at the following URL's that help in the understanding.