mod_comprofilermoderator

In Joomla in the module you can set hide, if empty. That doesn't work with mod_comprofilermoderator. Even without any content the title and the empty module is shown. Would be good, if that works. Could you check this? Or a message like "no action for the moderator" would be also good.

It already should be doing that. It won't output anything if there's nothing to output. This means if the pre-text and post-text parameters are empty and there's no links to display it won't display (either core links or from triggers) the module. What settings are you using within your module? If "CB Plugins integration" is set to "Yes" try setting it to "No" as maybe a plugin is preventing it from being empty.

I disabled CB Plugins integration (for which plugins is this? Which messages the moderator gets here?) and the empty module still shows the title. Enclosed my settings. What can I do?

Which layout are you using under Advanced Options tab? If using horizontal layout it maybe still outputting the span elements. Beyond that try right clicking and inspect element on the module and see what HTML it's outputting to cause it to still display. Even a linebreak in pre-text or post-text could be causing an output.

Good idea looking into the html output, unfortunately it is nothing beside the title

and

Could you please tell me:
CB Plugins integration
for which plugins is this?
which messages the moderator gets here?

Try switching to default Joomla protostar template. Seams like it's your template doing it maybe. The module doesn't output the title as that's actually the responsibility of the template. Remove the title and see if the module successfully hides as in my tests this all works fine using protostar template.

CB Plugins integration
for which plugins is this?
which messages the moderator gets here?

CB Gallery will if you've approval enabled.