Thanks @Citric for the answer.
My other questions I can now answer myself and maybe the answer is helpful for others new to this area. It seems not to be possible to flash the AGX with SDK Manager. Instead you really have to do it with the flash.sh script.
Following the description in Seeed wiki and Readme.txt in downloaded 5.0.1 driver package from Seeed (only the folders are a bit different) I was able to flash Jetpack 5.0.1 to the Xavier without any issues.