Depends on the program. Naturally, you want to invite your favorite candidates first, but it depends on when candidates submit all the required materials, so if your #3 submits first - they may be invited first. Generally, after Midyear you have your "Standout Star" (one or two), who you ideally want to wait for, your "Can do do the job" people - five or six, sometimes fewer depending on how complicated your position is, and then a bunch of people you wouldn't invite at all. Sometimes upon closer look exact opinions will change. For the "Stars", the competition between programs may be fierce so some programs will make offers to them before they have interviewed all the candidates... some programs will wait until they have interviewed everyone before extending an offer.
As far as preparation for on-site interviews, it would be very similar to Midyear... don't know if I would do anything different. I guess, I would look up a bit more information about the company, maybe look up the names of your interviewers (once you receive the agenda) to see if you have anything in common with them that you might bring up... Be prepared for the unexpected - someone may be running late, unable to meet with you, have a conversation on the phone instead, there may be a replacement interview, your flight might be delayed so you arrive at 2am and your interviews start at 8am...
And one piece of advice - be the best version of yourself you can put forward, but still be yourself. Otherwise, if you are hired and in the long run you turn out to be something entirely different (and no one can keep up an act for a year or two through all kinds of circumstances), it will not make your colleagues happy...
and freaking out ..
