I unable to find anything related to my question. Is there options in Directory Opus to display something like "Files copied successfully" or similar instead of just disappearing file copy progress indicator?
No, there is no summary option. The closest thing you can do, is to use the file log protocol option, which is turned on by the command Set Utility=Filelog, but unfortunately there is no file count summary. You can copy that code below, & paste it into a toolbar, while in customize mode, to add a file log button.
You can configure your log file in settings -> preferences -> file operations -> logfile.
<?xml version="1.0"?>
<button backcol="none" display="both" textcol="none">
<label>File Log</label>
<tip>toggles the file log utility </tip>
<icon1>#edit</icon1>
<function type="normal">
<instruction>Set Utility=Filelog,toggle</instruction>
</function>
</button>
Simply add this after the Copy command:
@confirm Copy finished|OK
Opus will then display a dialog box when the copy finishes.
Oh, cool. That option didn´t came to my mind. I always used the @confirm before other commands. 
Great! Thank you!