There are two main differences in DOM between valid HTML and XHTML served as text/html - in XHTML all elements are in lowercase and UAs don't create implicit elements such as tbody in XHTML. That's all! You don't need to put inline styles and scripts in CDATA sections. UAs handle these XHTML pages practically the same as they handle HTML ones, xml well-formedness is not checked. And of course, all document.* collections work, innerHTML works either. And yes, document.all works even in Firefox (since version 1.5).
That is the preferred MIME-type for XHTML pages. Commonly it is used with content-negotiation mechanism to serve pages as application/xhtml+xml for browsers that support it and to serve pages as text/html for browser that doesn't support application/xhtml+xml. Some people say that most of neat DOM methods and properties do not work in XHTML document served as application/xhtml+xml. They say:
document.write doesn't work
innerHTML doesn't work
document.applets, document.forms, document.anchors, document.images, document.links, other document.* collections don't work
document.all doesn't work
document.body and all other properties of document don't work
XHTML document served as application/xhtml+xml has the same DOM as XHMTL served as text/html.
At first Mozilla has been creating XMLDocument object for pages served as application/xhtml+xml, but then there was opened a Bug in December 2001, and the first fix was proposed in March 2003 and in two monthes final fix was accepted. Since that time Mozilla started creating HTMLDocument object for XHTML pages served as application/xhtml+xml and therefore they had to support all that HTML stuff: document.write, document.applets, HTMLElement.innerHTML, document.forms, document.anchors, document.images, document.links, document.cookie
The only thing they didn't support is document.all, it's only supported in quirks mode of document served as text/html.
As for Opera: it checks namespace of element to understand if it is xhtml and therefore there should be a HTML DOM or if it is a xml document and there should be XML DOM. And yes, Opera provides HTML DOM for ALL MIME types that XHTML can be served as.
Mozilla creates XMLDocument for XHTML served as application/xml or text/xml, so it doesn't provide HTML DOM for such documents. But Opera does.
Yes, that may sound strange, but it's a fact - innerHTML is supported everywhere, FOR EVERY XHTML MIME-TYPE.