Thank you for your answers.
I also found some time to try around a bit and I feel like I read probably every forum and thread about UR and Profinet by now, but I still couldn't get my UR3 to show up as a member in Profinet, and therefore never got the "LED" in the Profinet-Tab in Polyscope to switch to green.
Reason fot that and one thing I have never seen mentioned anywhere, was a step that might be obvious to you or maybe even to everyone but me, but I think it doesn't hurt if I post it here anyway: The UR has a configurable network address, that needs to be set under Mainpage-->Setup Robot-->Network (obviously the Profinet members need to be in the subnet, e.g: PLC: 192.168.0.1, UR: 192.168.0.2). If this Network-address is not set (you can check in the 'About the Robot' window) you cannot use any network connections with the Robot