You can submit your AMCAS primary (after they've received all of your transcripts) the first day they "open" in June. You need to check off at least one school to submit your primary. You can check off as many as you like, but some people like to wait and see what their score will be before committing (i.e., if they bomb the MCAT and want to apply next year instead, they would be considered a "reapplicant" and have to show significant improvement for any school to which they've already submitted their secondary - or, they just don't want to waste money applying to super reach schools if their MCAT doesn't come out as hoped). You can add more schools to your primary AT ANY TIME. You can add them later in June, or in July after your MCAT score comes out, or even later in the season. It's up to you.
After AMCAS sends your primaries out around June 21st or so, some schools will immediately send you a secondary. However, most send their secondaries out in July (with a couple in August). As soon as you get a secondary you can fill it out, return it, and have your LORs sent in. That way, the only thing you're missing is your MCAT. As soon as your MCAT is released, it's automatically sent to those schools you've checked off on your primary. You will be considered "complete" and available for review right away at those schools. Some schools, however (not most), screen, and they will wait to see your MCAT before sending you a secondary. You will be a little behind for those schools, but not by much - especially if you pre-write the secondary essay questions and can turn the secondary around asap once you get it.
Every school I've heard of will wait for your score before even taking a look at your application. Until you're complete, you can't be reviewed, so you won't be rejected until your MCAT is in.
As for DO schools, I have no idea how that works as I didn't apply to any.