Thanks for interest in this and trying to help.

cat /etc/sudoers | grep bnf
Result in:
Ho ever, there is some strange things I've noticed. Obviously, 'sudo bnf;sh' invoked from terminal works, but if I copy and rename (or any other) script, for example 'test-bnf.sh', I can:
successfully, even given fact, that there is *no* sudoers file for test-bnf.sh! Of course I'm sure this script requires root privileges, it doesn't work without sudo (it's basically copy of bnf.sh, with different name). Maybe it's default behavior, when rootsh package is installed?

I've also tried Your desktop file, but it doesn't make any difference - if i set
...(or any combination involving quotations, single or double), it does literally nothing. If i set
...(inclusion of & at the end doesn't seems to make difference for desktop file, but I've checked it anyway), it works, but first it show terminal window for a ~1,5 second, terminal disappear, and only after that notification pop-up is shown. so, it's quite slow for it's purpose. BTW, that's what I've called "flash" terminal window for a second.

At the same time, using ShortcutD with:
...result in it working as it should, i.e. pop-up notification appear immediately, without terminal window in-between (obviously, cause it doesn't call osso-xterm). Meanwhile, using queen beecon widget (snippet or beecon, doesn't matter) with exact same command (sudo bnf.sh), result in nothing, just like 'Exec=sudo bnf.sh' in .desktop file.

Yea, I can live with desktop shortcut that show terminal before pop-up and is quite slow, but I'm just curious why it looks like that. What I don't get is why 'sudo bnf.sh' work flawlessly from terminal and ShortcutD, but doesn't from .desktop file 'Exec=sudo bnf.sh', or other tools like Queen Beecon Widget or Desktop Execution Widget... Are .desktop files supposed to work like that (deny to work with 'sudo'), but ShortcutD do it different way (it says, that it's just invoke custom command, so it *shouldn't* be different from what QBW or DEW are doing...)?