Things like this don't "just happen", they happen for a reason.
As far as I know, Opera was installed on my computer when I bought it. Of course, if that's true, why did it suddenly begin auto-starting?
There's no malware on my computer. Maybe Opera piggybacks onto something else? I could have overlooked a tiny box that was already checked. Some programs try to sneak onto computers that way. But I hadn't installed anything recently, when this started happening.
I'm going to be very, very careful to avoid Opera in the future, but I'm normally quite careful already. I tend to be paranoid about internet security. So I have no idea what happened in this case.
I'm also hesitant to start screwing around with the registry, given that I know nothing about this stuff. I might just uninstall Carbonite and switch to a different backup service to see if that works. But we'll see. I certainly appreciate the comments here.