Your question is far away from subject line which is misleading. To check whether process is running you use ps or kill. But you are asking how to check what process is doing, not whether it is running.
Yup; GUI applications often have a status bar or other progress bar of
some sort indicating what they are doing. If not, perhaps log file.
Worst case you could us something like ‘lsof’ or ‘strace’ to watch a
process to see what it is doing with the system, but really that’s a bit
painful. If you’re in KDE a copy operation should have a status bar or
notifications in the notification application.