With LE 10 Nightly (Index of /) when the display is connected only to HDMI1 on RPi 4B, both audio & video works fine on it.
The same happens when the display is connected only to HDMI0.
When both HDMI0 and HDMI1 ports are occupied (by active HDMI devices), Kodi displays video on HDMI0 only and audio can be switched to any HDMI output.
I don't use VPN so I don't have personal experience and I did not have to struggle with DNS settings. But when I quickly look at your log & resolv configuration in my LE 10 Nightly, at first I would check the /storage/.kodi/userdata/addon_data/service.vpn.manager/update-resolv-conf.sh file, especially line 66 which needs to be modified somehow to fix the syntax error reported above in the log.
I suppose this script should perhaps modify the /run/connman/resolv.conf file, generated by Connection Manager by default, which contains a current DNS server's IP address. So you should update the script to do what is expected.
I am afraid nobody can tell you the release date of LE 10 stable. AFAIK, RPi 4B has a priority but a lot of things has changed from LE 9.2.6 and it will take some time to finish that hard work. My personal estimate is several months.
It's really strange. Do you have a customized bootloater config / did you also changed it when replacing bootloader versions? What's your current?
Are you sure the SD card / USB key imaging was done properly?
Also this is strange:
I tried the 10 beta, and it could boot properly. But it's too unstable (I'm getting no wired ethernet, CEC doesn't work reliably, reboot via menu gives a kernel panic...) so it's not a solution I can live with.
I am running LE 10 Nightly builds but never had an issue with wired ethernet nor kernel panic when rebooted from menu.
Didn't you try to overclock your RPi?
If not, I am out of ideas and perhaps there could be really a compatibility issue with RPi 4B v1.4 but I am afraid it won't be ever analyzed in LE 9.2.6. Move to LE 10 Nightly builds and you won't be sorry. Despite there are some issues, in general it works fine and the progress is every day.
Make sure you are using HDMI0 output (next to the power connector).
Also try with another HDMI device (a PC display with FHD resolution if possible) to test if there's any change.
No problem. 
RPi 4B is a fair-priced, great piece of HW with many use cases for a long time due to excellent support from vendor & community.
But using RPi as desktop - it depends on RAM size you have. I own two RPi 4B but only 1 GB and 2 GB models. Both are not very suitable for a "full" desktop so at least 4 GB model would be much better.
In every case - a boot from some SSD HDD connected through USB3/SATA adapter instead of SD card is necessary.
With a good power supply and with a good cooling (there are several great ALU cases) it can be running permanently for sure.
Do not underestimate the required power supply performance, otherwise you will notice strange functionality problems that can result in data loss.
RPi 4B itself in fact needs 2x more current than you expect.
Furthermore, in the case of rotary HDDs, it is necessary to take into account the peak power consumption at power-up, which is usually above the (quite low) limits of the RPI 4B USB ports. An USB "fork" cable (with a power plug, which can be easily made) or an USB hub with external power is then a must.
I have no experience with DVB-C but Astrometa tuner (15f4:0131) works fine on RPi 4B-2GB / LE 10 Nightly builds / Tvheadend Server 4.2-4.3 / Tvheadend HTSP Client / DVB-T2 so I believe it should work with DVB-C as well.
Reboot your RPi, connect to LE through SSH console, copy & paste and put here output of:
lsusb > /tmp/info.txt
echo >> /tmp/info.txt
dmesg | grep DVB >> /tmp/info.txt
echo >> /tmp/info.txt
ls /lib/firmware/dvb-demod-* >> /tmp/info.txt
echo >> /tmp/info.txt
ls /storage/.config/modprobe.d/ >> /tmp/info.txt
echo >> /tmp/info.txt
ls /dev/dvb/ >> /tmp/info.txt
paste /tmp/info.txt
#Unfortunately it's not possible here due to different drivers used in latest LE version.
Post 2 does not work, an analog sound is not offered under System.
Of course it works in LE 10 (tested on latest Nightly build) if you edit the /flash/config.txt properly and set the Audio output device in Settings - System - Audio menu to ALSA:bcm2835 Headphones, bcm2835 Headphones
Well, the USB controller which RPi 4B has it's not the one of best ones regarding to compatibility with older USB devices - I have an old USB HUB, working everywhere except RPi 4B.
The latest bootloader and VL805 USB firmware releases are here: Releases · raspberrypi/rpi-eeprom · GitHub
The bootloader Release Notes: rpi-eeprom/release-notes.md at master · raspberrypi/rpi-eeprom · GitHub
And the firmware itself (which is located on filesystem): GitHub - raspberrypi/firmware: This repository contains pre-compiled binaries of the current Raspberry Pi kernel and modules, userspace libraries, and bootloader/GPU firmware.
Mount the drive to Windows and run dskchk.exe to clear the problems, then eject the drive properly.
Just for clarification - the command in Windows is chkdsk [driveletter:] /F
I have downloaded the .gz image (full install) rather than the .tar image (update).
Yes you are right, sorry for the wrong link.
Can you also try to write the LE image to USB stick and try to boot from it if it's the same issue?
Try booting RPi+LibreELEC on a (non-4K) PC monitor or other TV to see if the basics work
A couple months ago I experienced LE (yet installed) couldn't boot with some older AOC PC display (with DVI-D and VGA inputs only) connected through HDMI/DVI-D adapter. As this display had also some strange issue with PC connected through DVI-D (it was not possible to set the resolution to 1920x1080), I did not try to fix / analyze that and trashed it.
But maybe in this case I would try to boot RPi without any display attached to see if there's any progress at least with partition resizing.
A write-protected SD card usually means it's dead and unusable. It's not unusual.
Perhaps you may try the latest LE 10 Nightly build: LibreELEC-RPi4.arm-10.0-nightly-20210505-83202cd.tar
Make sure you have the HDMI cable connected to HDMI0 port on your RPi 4B - it's next to the power connector.
