There's nothing that could normally explain this. What it sounds like, is that PS somehow assigns the dot gain profile to an RGB, three-channel image, which of course will return an "invalid" message, because it is. Normally, such an operation is not possible.
So it sounds like a corrupt preferences file, which can cause all kinds of erratic behavior. Under Edit > Preferences > General, press "reset preferences on quit". This returns Photoshop to its out-of-the-box factory state.
Save out !!! any custom actions, brushes and so on first. They are stored in the prefs file and will get deleted.
Thank you D Fosse for answering!
However, I have managed to pinpoint the issue, and it is not in Photoshop, as in turns out. I would upload the photos in question to a website and later download them, and the downloaded copies would be the ones saved with a Dot Gain 20% profile and in Greyscale mode, not the ones converted by the Image Processor. As it turns out, it is the CMS handling the conversion of the photos after uploading that causes the problem, for some reason it makes those black-and-white photos get a Dot Gain 20% profile and Greyscale mode. Opening them in Photoshop and changing the color mode to RGB automatically brings their sRGB profile back. The issue is in the CMS, luckily!