I tried to use some http://www.freepdfconvert.com in order to convert a PDF file created by Prince to a Word document (The reason I don't convert directly from HTML is because of the counters, which are not supported in my browsers, so they don't get copied to Word).

I got a DOC file, but it seems that the main part of the document is included in a picture or in a text-area, and can't be edited.

Do you have any idea, what I can do so that my PDF document will be convertible to DOC format?

Perhaps try "Save as..." from Acrobat 7 Professional, which offers the options of exporting to RTF or DOC format. But make sure that you use the latest revision of Prince (which is Prince 5.1 rev 9), as we have just included a workaround for a bug in the RTF export.