This works fine if there are dead cells in the image, however, those are very rare, and therefore I'm getting random signals identified (there are no dead cells) on the other images.
You should check your module outputs to be sure, but if as I suspect the images where there are NO dead cells have a very low threshold in IdentifyPrimaryObjects (like 0.01) compared to ones where there really are dead cells (like 0.2), you should be able to set an threshold minimum so that only "real" dead cells are called.
After having identified those dead cells, I would like to exclude those objects from the further analysis (preferably automatically for all channels). Is there a way to do this (e.g. get the object ids and exclude them from further analysis)?
Yes! You can use the FilterObjects or MaskObjects module to exclude the nuclei from dead cells- which one you will choose depends on the details of your pipeline, so I recommend reading the help for both to decide which one is right for you.