This closely resembles your other thread "What do you think"... but I realize not all your questions were answered.
Yes, it is definitely to your benefit to have your application in early----as early as Jun.1 is not that important.
Speaking from my personal experience---AADS received my internet application---immed. I paid my application fee through the internet and fed-ex my transcripts, personal statement, etc.... overnight delivery. They recieved everything on 10/04 and it was processed and sent back to me-- and the schools 10/11. I received them-- as I'm sure my schools did-- on 10/15. The DAT scores are sent immed. after taking your exam. (Everything is electronic and you also get your scores immed.)
So based on my experience and banking on no problems----it takes ~10 days after your application is received to be process and sent to the schools. I think the DAT scores depending on how far away some of your schools you applied to----shouldn't take but 4 days to be received. Therefore you could send in all your application stuff and within 4-5 days after you could take the DAT and everything will wind-up together
(close at least---barring no problems)
If you are dead-set on getting that applic. in and DAT finished by Jun. 1---great! If you find it is too much and your not finding time to dedicate towards the DAT--- by all means take a couple of weeks in late May or early June to study. Even if you get everything in by late june------You are still way a head of 98% of the applicants and will enjoy every benefit afforded to the early bird.
Hope this helps----once again this is how it worked for me. Good luck
R.R.B