I create a new project using the new project wizard so that I get unique project GUIDs assigned. Then I would use "Add Existing Item" to copy items from similar projects if it made sense to do so.
I sometimes use a file diff tool to copy references from one project to another, otherwise I just add the references by hand. A file diff tool can also be used to include similar source files, but the underlying files have to be copied anyway, so I prefer "Add Existing Item".
I've used T4 to generate solution and project files, but that definitely seems like an edge case and not something that would normally be necessary. In that case, I'd probably wrap the T4 in a PowerShell like script to create and populate the rest of the directory structure.
I use "shared libraries" pretty aggressively in general, but not specifically due to this scenario.
In general, I don't find myself reusing plumbing between projects much. It's probably more often that I hack away in one "prototype" project, then abandon it, and rebuild the project from scratch following the above approach and only bring over the "non-hacky" code.