Have you tried connecting a charger with PD to the USB-C port? Maybe the switch doesn't supply enough power for the display.
Someone else seems to have managed it here on YouTube (with a different monitor that is also powered separately)
I hope this helps.
https://www.youtube.com/watch?v=N_uWCYIjj3o&t=2s
PS. As usual - you do such experiments at your own risk. 😉
Hello Kongsberg. The device must support DP Alt Mode. Technically speaking, this is a DisplayPort via USB-C. This is not always clearly defined - on many devices there is either nothing at all or it is written somewhere in very small print whether video is possible via USB-C. This was especially the case with the cheaper models I encountered (e.g. HP Pavilion). In short: USB-C is only the connection - the technology behind it must support it, otherwise it doesn't work. See also here: https://www.club-3d.com/de/technology/15/usb_c_over_alt_mode/
By the way, I also stumbled across this at the beginning of my USB-C time (DP Alt Mode, PD etc.). - all wonderfully poorly documented or not documented at all...) so I understand the frustration.... 😉
I can't answer the first question, I don't have an HDMI to USB adapter. (Do they even exist in this direction? USB-C -> HDMI yes, but HDMI -> USB, I have never seen).
Second question, yes that works. I'm currently using mine as a 3 monitor on a docking station. (NB, Monitor, HP E14.