# means the image is untagged, i.e., Photoshop doesn’t know what profile to use because the image was saved without profile information, and you have told Photoshop to not color-manage these images.
* means that the document colorspace is different from your working colorspace. This means the image had an embedded profile tag BUT that you have Photoshop set not to convert images to the working profile upon opening them this is usually good. The * is just a reminder, the image should look “right” as long as you have your proofing set up properly.
/CMYK or whatever at the end means that the indicated proofing mode is being used i.e. Logo, RGB/8/CMYK is using your preferred CMYK profile for proofing, so you can see how the image might look if, in this case, printed.
I think some of the problems you’re having come from having Photoshop open an image with no profile info and either letting it convert the image to a different profile, or letting it assign a profile and tagging the image when it’s re-saved.
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