Is it possible to make renaming automatically?

I have a folder that I sort manually. Is there a way to make it rename files (from 0001 to infinite) when i change my sorting order?

If it is not possible to make it automatically, how to make it manually? (If possible, by bypassing names that start with "_" (underscore)? )

The Advanced Rename Dialog provides sequential numbering, but I don't think there is an event for manual file sorting that could trigger a rename command.

There is a column Index available for the File Display, which you might like.