> What do you do when you want to update state in other places in response to a change in a component?

In 90% of the cases one of the parent components. Sometimes redux when the component tree goes super deep, or for things such as currentUser.

> How about if you want to make your component's state persistent (particularly when the component is external to your project)

What do you mean? Give me an example.

> Also, what do you do to generate state dumps for error reporting?

I don't generate state dumps for error reporting.