Nothing happens when Perform cut button is clicked.

[stiv-yakovenko]

I see task panel being opened, new task appears there, but I don't see any error nor any success message, I don't know where to get cropped file, tried to find something in the same folder, where original file is stored, but no luck.

[bartekmotyl]

Do you see the red background in the task row? This would indicate an error. See for example these screenshots - I provoked an error trying to save output file to a read only directory. Note you have to scroll far to the right to see the error message.
Please also note that by default output file is indeed saved in the same directory as input file, with name following the pattern {FileDate}-{FileNameWithoutExtension}.{Timestamp}{FileExtension}