I think that I've read all the FAQs but couldn't see the answer to this:

When I clone a template set, where does it store the clone? Is it stored physically on disk, or is it just stored in the database?

Secondly: If I later install a new module, what happens to its templates? Do they get added to the default template set or the new template set or both?

Thirdly: Is there a way to apply global changes to all the templates in a set? I would like to add class='formButton' to every form button on my site so that they are displayed consistently via the formButton declaration in my css.

I'm sorry if this is already answered somewhere, but I couldn't find it.

The templates are stored in the database.

If you add a module later you need to generate those for your customized template set. Just go into template management and generate them there.