I can run applications like Firefox with the application launcher, but my scripts in ~/bin do not appear there. Also I would like to run functions from my bashrc from the plasma GUI. Is that possible?

You could create some *.desktop files in ~/.local/share/applications pointing to the scripts. Kmenuedit or dolphin's context menu: Create New>Link to Application... can do it conveniently or just make them manually.

Thanks, that helped!