So, in a nutshell, yes, Trados 2006 is compatible with OO, it just doesn't have an OO interface like the Word interface. And you CAN translate Word files without having Word on your computer, provided you use TagEditor.
I haven't checked but I'm nearly sure that you need MS Word installed to import Word files into TagEditor.
The second method proposed by Viktoria (convert Word file to OpenOffice and then translation with TagEditor should work).
What I don't like about translating Word files without having Word installed is that you cannot open the file with Word before sending it back to the customer, so you don't really know if the file you are sending is OK or if your customer is going to find problems when they try to open it with MS Word...