Is there a general rule, when one should use document.write to change the website content and when to use .innerHTML?
So far my rules were:
innerHTML and document.write are not really comparable methods to dynamically change/insert content, since their usage is different and for different purposes.
document.write should be tied to specific use cases. When a page has been loaded and the DOM is ready you cannot use that method anymore. That's why is generally most used in conditional statements in which you can use it to syncronously load external javascript file (javascript libraries), including blocks (e.g. when you load jQuery from the CDN in HTML5 Boilerplate).
What you read about this method and XHTML is true when the page is served along with the application/xhtml+xml mime type: From w3.org
document.write (like document.writeln) does not work in XHTML documents (you'll get a "Operation is not supported" (NS_ERROR_DOM_NOT_SUPPORTED_ERR) error on the error console). This is the case if opening a local file with a .xhtml file extension or for any document served with an application/xhtml+xml MIME type
Another difference between these approaches is related on insertion node: when you use .innerHTML method you can choose where to append the content, while using document.write the insertion node is always the part of document in which this method was used.