Given your performance the first time you took the MCAT, you should definitely prioritize studying for the test over completing the application. If your apply a bit later but score well (30+) on your second MCAT, you'll get looked at. If you prioritize finishing the application over studying for the MCAT, and get a 23 again, even if you submitted your primary application in the first day of June, your chances would be pretty low. That being said, I wouldn't prioritize the application so low that you don't finish it until the end of August - that'll definitely reduce your odds.
I would initially submit the application before receiving your second MCAT score (especially if you put it on the back burner while you study for the MCAT), because it'll save you tons of time. Like I mentioned above, if you're not confident in your score, just put one school on the application initially. Depending on when exactly you submit for verification, it's possible that your MCAT will get scored before your application even gets verified - so schools would get both scores at the same time. Even if they don't get the second score right away, it won't matter as long as you tell them you retook it. And yes, AMCAS automatically loads any scores for existing MCATs
On the issue of postponing your MCAT - it's hard to say. If you think your score will improve significantly by delaying the test 1 month, then it's probably worth it.