My website (created with CC HTML Editor 2010) works fine on computer internet connections, but when I look at it on my BlackBerry, it shows up as nothing but pages (and pages) of oriental characters.

In a possibly related note, an automated website evaluation tool advised that I should stop using "unicode" - yet that seems to be the default for CC HTML Ed. Do I need to change this somehow?

Just choose different character sets and encodings etc. You could probably just make yourself a snippet ready to pull out every time you set up a new htm document. The doc types can be set up in preferences, but I think these other encodings will need to be set separately
http://www.w3.org/International/tutorials/tutorial-char-enc/#Slide0080

Instead of "unicode" or "utf-8" try "iso-8859-1".