Vlad
I gave a few suggestions how the icons could look like above, for example: a rocket with a star, space station, galaxy, a different planet (Earth, Mars, Neptune with a star in the middle perhaps)
I tried a Mars icon from this pack and it looks cool, maybe something like that, but with Orion's star in the middle: https://findicons.com/pack/475/solar_system (example attached)
- or allow user to set their own icon, because right now it overrides the custom dock icon on launch if you change it in info.plist or replace AppIcon.icns.
I imagine it would either appear under the same menu as name of the profile and badge, or in the Settings > Appearance, where you can already change the app icon. More options or an alternative to import a custom icns icon. For the badge you could add an option "without" in the dialog with the profile name and badge.
Currently I don't see an option to remove a badge, you can only change it. I would expect one of the options to be none, or a white circle with red slash through it.