Is there a general rule, when one should use
innerHTML can be used to change the contents of the DOM by string munging. So if you wanted to add a paragraph with some text at the end of a selected element you could so something like
document.getElementById( 'some-id' ).innerHTML += '<p>here is some text</p>'
Though I'd suggest using as much DOM manipulation specific API as possible (e.g.
<element>.appendChild, etc.). But that's just my preference.
The only time I've seen applicable use of
document.write is in the HTML5 Boilerplate (look at how it checks if jQuery was loaded properly). Other than that, I would stay away from it.