Is it possible to do a ALL SERVER DML Trigger?
You are not going to be able to create a server-wide trigger, because it has to exist in the same database that the DML changes are occurring in.

Having self-deleting code sounds a bit problematic as well.

I think you are going to have to have a process in place that creates the triggers when a new database is attached, and removes them when you are finished with them.  This is likely going to be a manual process.