Is it possible for me to change both the properties & the HTML of a content editor web part on a publishing page using SharePoint 2010's web services? If so, would you mind pointing me in the right direction?
Yes you can.
The contents of a Content Editor Web Part (CEWP) are stored like any other web part - in the web part properties.
You can see this if you Export the web parts .dwp file and open in a text editor, you will get something like
The contents are in the
You can read and write the web part properties using the WebPartPagesWebService - see GetWebPart and SaveWebPart
Actually YES (thanks Ryan)!