Import your favorites into IE. Then all your favorites will be stored in your user folder on the C drive and in that folder, there will be another folder called "favorites". Just save that and you'll be fine.
Honestly, if this were happening to me, I'd just uninstall Opera and then download the latest version and try to install it. If that didn't work, I'd format. There's obviously something seriously wrong and instead of spending 3+ hours trying to solve it, I might as well just format the sucker and fix that problem along with any other one's I had, while speeding the computer up considerably in the process!
To get a computer working and make it STAY working, just format it, disable the virtual memory paging file, turn off drive indexing, install AVG & Spybot, and use Firefox. That's pretty much all I do and I have a PERFECTLY working computer all the time. I format every 6-12 months, but it doesn't matter because all my stuff is on external hard drives or on my server. If my master computer totally crashed, it wouldn't phase me a bit. Just back your stuff up regularly, set up your computer properly, and format it when you run into a major problem.
But one thing you shouldn't do is use IE for any reason (unless you're a web designer and need to test usability). IE is the most common web browser because it comes preinstalled on ALL new PC's. Because of this, it's also the most vulnerable browser to attacks and malicious software. Back when I used IE (this was a loooooooooong time ago), I would get tons of crap I never get nowadays. I'd go to eBay and get a freaking virus or a ton of spyware! IE just isn't a good idea. I'm sure Chrome or Opera are viable alternatives, but I always just use Firefox and it's always kept me safe.