I had plenty problems getting it done and my initial post, where I was asking for help, never showed up here. It got marked as spam and I dont know why. Anyways, I try to share what I have learned since I have finally solved the problem.
Installing the ROCr support was not possible due to dependency errors and incompatible kernel version.
The command
sudo ./amdgpu-install --opencl=rocr --headless
failed. The ROCr depends on the amdgpu-pro for some reasons, although I think I dont need nor want to install the OpenGL components. (I am happy with the linux drivers that comes with the kernel.) Fact is that the "--no-dkms" flag don't help. So I need to make it work. For some reasons the libc6 on my system is version 2.31-0ubuntu9.3, but the amdgpu-dkms needs 2.31-0ubuntu9.2. I hope the best and downgrade the lib with the command:
sudo apt install libc-bin=2.31-0ubuntu9.2 libc-dev-bin=2.31-0ubuntu9.2 libc6:amd64=2.31-0ubuntu9.2 libc6:i386=2.31-0ubuntu9.2 libc6-dbg:amd64=2.31-0ubuntu9.2 libc6-dev:amd64=2.31-0ubuntu9.2 libc6-i386=2.31-0ubuntu9.2 locales=2.31-0ubuntu9.2
I am using kernel 5.14.9-051409-generic and so I cannot build the driver on this kernel, so I need to reboot the system. (Make sure to run amdgpu-uninstall bevor reboot, when installation try has failed!) Hit ESC on boot to enter grub menu and boot with kernel kernel 5.4.0-88-generic.
Now, the install via command
sudo ./amdgpu-install --opencl=rocr --headless
works. Reboot again and don't forget to add the user to the groups "render" and "video". Then start blender (2.93.4) and voila, OpenCL device navi 10 RX 5700 XT is available. On the first render try blender chrashes and I get afraid... reboot again with kernel 5.4.0-88-generic. This time, I can render with cycles and GPU sucessfully.
Reboot with 5.14.9-051409-generic and still everything works fine. I can render on GPU, CPU and both. I am exited how the rendering times will be and start a quick test.
I set up a simple test-scene with some lights and cubes and rise the samples up to 512 samples.
These are the results of rendering the same frame on GPU, CPU and both at once:
navi 10 RX 5700 XT
66,89 sec
Ryzen 9 3900X 12-Core
51,43 sec
navi 10 RX 5700 XT AND Ryzen 9 3900X 12-Core
30,39 sec
It looks like I have a working setup. So now it's time to close the terminal and get all the things done in blender that were on hold for so long....
Best regards!