I have never used the free version of the HTML Editor, but according to the specs, you can use HTML5, which is the current standard. The programme doesn't update anything for you, you need to do that yourself. That is one of the best features with all versions of the Editor, that you are in control of things yourself.
Manually changing from some html4 version to html5 is easy, you just change out the doctype and the charset declaration from the old blurb to the html5 ones. In the settings you should also find the options for new documents, where you can select html5.
I still have some old sites written in html4.01, and they work fine both in old browsers and the newest ones, so I don't quite understand your problem with that.