The best solution we could come up with to meet all requirements involves the following steps:
I realize this looks like a lot of work, but it shouldn't take more than 30 minutes to set up. It works great and will save development effort and time in the long run:
Visual Studio makes this easy: drag a TypeScript file (either individual files or a _references.ts file) from a shared project to a referencing project's _references.ts file or to any other TypeScript files (_references.ts can keep references nice and neat, but it might not benefit all project structures). Note that the _references.ts file must always be in the root of your project in order to work as expected.
This step ensures that shared references point to the original shared files using a method that works with Visual Studio, Chrome, and Firefox debugging (replace relative shared references with Project URL or Virtual Directory).
To Microsoft: This type of referencing should be configurable and/or automated in a future Visual Studio release, e.g. if a file is referenced from a different project, automatically map the reference to the shared project's Project URL.
This works for multiple debugging systems (tested with Visual Studio, Chrome, and Firefox):
Make sure you set up your shared Web App project for debugging along with your referencing Web App projects.
This does not work for Visual Studio debugging, but might be a better option for some scenarios, and it's the only known option for non-web application projects (tested with Chrome, probably works for Firefox):
Powershell [io.file]::WriteAllText('$(ProjectDir)Scripts\App.js.map', ((gc '$(ProjectDir)Scripts\App.js.map') -replace '../../GlobalRef/','/GlobalRef/'))
If you need to be able to debug TypeScript files in the production environment (not one of our requirements), then you could alter the post-build script to run a different command (similar, but point to your shared website's Virtual Directory instead of localhost). That's outside the scope of this post.
That is by far the best method I could come up with (thanks to all of those other sources) that meets all of the requirements.
Please let us know if you have a better method.