Yes, the app opens in early May and you can start typing stuff in there. Also, you can go ahead and get transcripts sent in May.
Nope, you can't submit until early June. What that person may have been thinking of is what happened to me last year. I had all my transcripts in well before June 1st. I was surprised but AACOMAS actually verified my grades before June. So, on on the date around June 1st that submissions began I submitted my primary and it got to the school super fast!