All screens go blank when the PC is plugged to a docking station

This is a known bug in the kernel with DP over USB, which affected me recently, see here. A patch is available but does not seem to have reached the mainline Ubuntu repositories yet.

For me, the easiest and probably most stable immediate solution was to instead use the older kernel 5.4, which Ubuntu 20.04 supports out of the box and should be supported until the end of life of 20.04, and for which the bug was not there. See here for explanation on how to boot your system into an older kernel. Unless you have specific use for the newest features of the 5.11 kernel, I would try this first.