from my experience, most errors that appear at an epub check are caused by the basic html file, not the css or something similar. Even if you convert a document with a program like calibe, this errors will remain.
Here is how I do it: I convert all the documents to html (when it comes to word, just export or save it as html). Then I edit this html file with an editor like Notepad ++. Once I finished it, I use an html Validator to check if the html file is xhtml 1.1 valid. I recommend Total Validator
http://www.totalvalidator.com/ - it is free and you can download a small offline version, so you don't have to be online to check.
The Total Validator will give you the results in your browser (I guess this is similar to the epub check you used). Just make sure to set it to xhtml 1.1.
In the beginning, I had a truckload full of errors. When I began with epub, I was a total novice with html code, so I used the opening p tag without a closing p tag and the other way round ... and just to add this: just because of this errors, the final epub looked weired. Sometimes, lines were cut off on the border of the display. With a valid xhtml 1.1 file, this will not happen