Your application is complete when the schools has EVERYTHING. There are a few schools listed in that ADEA book that will begin reviewing your application before receiving you DAT scores. A few schools may start reviewing your application if you send them a copy of your DAT scores, but may or may not make a decision regarding an interview until they receive them electronically. You should definitely call each school before sending your scores in yourself, because some explicitly say on there website (usually FAQ) that you should NOT send your scores.
There are also some schools that will start reviewing your application before LOR are received. But majority of schools require EVERYTHING in order to be listed as complete.
I submitted my application a few days after it opened. I took the DAT on June 30, and my scores were received by schools on July 18. Could take a bit longer sometimes. I still got a lot of interviews even though my DAT wasn't the earliest possible.
To be safe, you should always write the DENT PIN and application number on materials. Most schools ask that for application fees as well.