Great thanks to everyone who replied!
Actually, not every AppImage is broken. For example, Picocrypt and Standard Notes are working, while Etcher and BetterDiscord are not working. Looks like only Electron-based apps affected. But I do not know, is problem in my local system or in some recent change by related packages. Can you please test apps mentioned above and tell me if it works for you?
--disable-gpu-sandbox option is really helps, it lets apps start and work normally without GPU process isn't usable error. But without this option apps still not working for me.
I got it about --nodeps Pacman’s option, I shouldn’t have used it, but sadly I did. Thank you all for warnings. Intuitively I thought that since I mentioned particular package in this command (pacman -Syu --nodeps boost-libs) Pacman should make exception only for this one package as dependency for any other packages, but not exception for every single dependencies packages during upgrade process. I’ll be very grateful if someone explain me how this command really works and what it actually does. It would be useful to know.
Since I thought this way, I also thought that this action can break only my problematic package from AUR, and even if it will, I always can just reinstall both with clean build option to make it work again. Was I wrong in these hypotheses?
Just like you suggested, I manually got fresh up-to-date mirrors from Arch’s Pacman Mirrorlist Generator for my favorite location and paste it to my /etc/pacman.d/mirrorlist file. Then I executed sudo pacman -Syyu, and yay -Syu after reboot. Actually my system is pretty up-to-date at the moment since Pacman found only one updated package (just one from qt5-* family) and Yay not found any AUR updates.
Pacman & yay don’t show me any errors or warnings, only very usual messages during upgrade. As I mentioned in my first post, Yay only tells this: ceph-libs: local (15.2.17-1) is newer than AUR (15.2.14-6). There is no such package in official Arch repositories, but it is present in AUR, and it is a dependency for boost-libs package, which is in turn a dependency for problematic synthingtray package from AUR.
Now I need your advice. Should I downgrade ceph-libs package to current AUR version with Yay, then forcefully reinstall boost-libs package with Pacman, and then clean build syncthingtray & rebuild/reinstall all packages which require boost-libs as dependency? Is it right thing to do?
Great thanks again for your help!