What you are looking for is a watched folder. Directory Opus does not do this. This post explains an alternative. I use watched folders via a program called Quick Macros. Multiple programs can do the watched part of what you want. Quick Macros has a free trial to try it.
As for the script part, Directory Opus can definitely do that. I wonder if you could just have a *.dcf file sitting in that folder that triggers to run when a change is made.
Could you explain in more detail what you mean about
But then, if there's a hole in the numbering (ie skip a number), it would always take the last file's number and add 1 to it.
Are you just saying you want the numbering to be one higher than the then the last file?