Where can I see output of an application/script (stdout, stderr) when I run it from KDE? This includes commands run from KRunner and *.desktop runs. What about running apps/binaries/scripts from Krusader? Does it delegates this to KDE or does it on his own? If it does not delegate this to KDE then again where can I find the output of these programs?

Probably /dev/null, i.e. the output is discarded.

If you want to capture the output there are a couple of options:
If you just want to display the output live as the program runs then you need to launch from something which will capture the output e.g. konsole or other terminal emulator. There are also some plasmoids like STDOUT Plasmoid https://opendesktop.org/content/show.php?content=161141 that will run a given command at a specified interval, displaying the resulting output.
You can set up a shell script to launch your app and redirect the stdout (and stderr) to files of your choosing.