Typically in the hotel industry, night audit processes "no shows" (people who haven't arrived yet) during the overnight hours prior to 6-7am, which is generally the time its considered the "next day" for them. If you're arriving after that time, you'd normally run the risk of being processed as a no show but there are often instances where people keep in touch with the hotel to let them know they are still coming or whatever their case might be. In Disney's case, computers roll over to the next day at 6am but with their online check in process, I'd say you could note an "after midnight" arrival. I would also call to have it noted on your reservation that you plan to arrive early in the morning on the next business day so that they know when to expect you.
With online check in, you can note your approximate arrival time and opt for a room that's ready as early as possible which they will prioritize over giving you a room that meets any requests you may have. Disney will try to align room assignments with arrival times, but its not always possible. It solely depends on how busy they are, how quickly housekeeping is turning rooms over, etc. Definitely a 50/50 chance and OP wouldn't know until they arrived.
Sure, do that. And you might not get your villa until after 4PM (which is check in time) or hotel room after 3PM (check in time). Book the night before, so you can hop right into bed when you arrive. Just add to your reservation that you are arriving late, around 7AM and will need the room at that time.