I found a solution and am posting it here in case somebody else finds him- or herself in the same troublesome situation as me.
The problem seems to have been caused by my having had EndNote X2 installed prior to X3 and then upgraded from Vista Business SP2 to Windows 7 Professional. Somewhere something went completely wrong, installing the CWYW dotm’s a total of three times in Word 2007. This made Word crash upon exit every time and in the logs I found that the problem had to do with wwlib.dll, i.e. the Word add-ins. It was obvious after some investigation that it was the EndNote add-ins (CWYW).
I uninstalled X3 and noticed that this simply restored X2 (!) whereas the program was not registered with Windows - only the file folders under /Program Files were restored. So I couldn’t uninstall X2 since it wasn’t installed… I found the old installation files and reinstalled X2 in order to update and restore all registry keys that are needed for X2, so that I would be able to uninstall X2 as well and then, hopefully, install X3 and get back to normal.
Installing X2 resulted in the add-ins in Word being updated to X2, but there were still six of them (3x2, EndNote cwyw.dotm and EndNote Web cwyw.dotm three times each). Uninstall went okay, but it didn’t solve my problem with Word - it still crashed, and it still believed I was using EndNote add-ins. For some reason it was able to activate these add-ins every time I started Word even though the files were no longer there…!
Then I found a solution on http://www.lazynetworkadmin.com/knowledgebase-mainmenu-6/2-windows/119-wwlibdll-or-add-in-crashing-word-2007, prescribing that I hack the registry a bit:
1. Close Word
2. Open Regedit
3. Go to HKEY_CURRENT_USER\Software\Microsoft\Office\12.0\Word\Data
4. Delete the Settings Key and any other keys EXCEPT “Toolbars” and “SettingsWordMail” Leave these two alone.
5. Go to HKEY_CURRENT_USER\Software\Microsoft\Office\12.0\Word\File MRU
6. Delete any instance of the offending add-in
7. Done, restart Word and the problem should be gone.
Doing this actually reset Word and it did no longer load the EndNote add-ins. And I could easily install X3 and - surprise! - I got all the functionality back, including the right-click menu options.