Thank you very much for your suggestions and making the screenshots and for the good tip regarding the key combo in the control panel. Power Adapter, switch is correct, using a USB 3.1 port in USB 2.0 mode.
It works now! I removed all “old” drivers - I enabled the “show hidden devices” checkbox in the Windows device manager and removed all old stuff (everything grey, not only related to sound hardware), removed the ASIO4ALL drivers, removed the UR44C driver, restarted the PC, disabled all onboard stuff I do not use and the onboard sound, disconnected the UR44C, booted up, installed the UR44C package, reboot, connected UR44C.
Windows Sound Playback works and I can now activate the UR44C Asio driver in Guitar Rig and in Cubase.
What still does not work: if I switch to USB 3.1 mode, Guitar Rig does not find the interface any more, although it is connected to a USB 3.1 port.
At least, I can use it and will now have a lot of questions on how to operate it properly 
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Thank you very much, again!