I think you can edit each field to set whether or not it is a default field. I believe that will impact whether or not it shows up in the list. Unfortunately, this may also affect other parts of the interface, and there is no direct way to manipulate the contents of this lookup.
This isn't a search I created. It is the "generic" search that comes up when I am trying to link a ticket to a global ticket... Does that make sense?
Yes, it makes sense, I inferred that from the screen shot. There is no direct way to control the columns or their order.
However, I believe that unmarking fields as default in the field editor for the record definition will remove them from the list. Please test this on one field. However, this may also affect other parts of the interface.