In answer to your question about what to back up, its actually really easy.
You're on Windows Vista, which is like Windows 7 (which, I'm running).
I would make sure you make a backup of the contents of the following directories
Note: xxx = your username
Bookmarks (Favorites) from IE
Desktop (in case you have items other then shortcuts stored there)
That should be about it. Unless you have a local e-mail client (Outlook, Thunderbird, etc) that you would have to worry about in regards to a local copy of your e-mail and or address book.
If this was a one time deal, I would copy all that info to a directory somewhere, so you could figure out the total size. Then, either burn that to DVD(s), or copy to a USB Flash drive.
Hopefully it does not come down to you re-installing Windows Vista to fix your problem with the upgrade of IE. I've been reading this thread, but have not been paying that close attention to your specific issue (wanted to give you some advise on what to backup).
So, I'll go back and re-read it all, and see if there is anything I can suggest that has not already been suggested.
Good luck! EDIT:
I did some looking, and I found this KB article on Microsoft's site
Most of the things listed there you have done already, and have not worked. However, Method #6 may be something to try.
Method 6: Run the Fix it solution to automatically diagnose and fix common problems with Windows Update
It looks like you can click a button and it will run some kind of check on your system to figure out what is wrong.
Normally, I'm not one to have a web site "Run a check and fix problems", but since this is directly on Microsoft's site, I feel a little safer running something like that.
So, give that a try, and see what it reports.
Finally, when you do try to install IE8 or IE9, do you get any kind of error message on the screen? What about any items in the event logs (Errors, Warning, Critical)?