My CI2+ worked fine with Cubase 6.5 under Windows 7, without any need for customization of any settings. The problems started under Windows 8. I reverted back to Windows 7 over the weekend, and it seems my problems are gone (at least for now).
I’ll have to do some more extensive testing, but during a quick test with a project that gave me trouble before I couldn’t reproduce the error.
It might have been a problem with some unsupported or bad drivers for my hardware on Windows 8, or just a general compatibility issue, I can’t tell.
For now, my problem is solved, but if I should ever run into the same kind of problem again, I’ll make sure to try @filterfreak’s suggestions (thanks, by the way)