Hi there! I was (and currently am) abroad for the entire application season. I had a very successful cycle as well. Here's how I did it:
1) I pre-wrote EVERYTHING and turned in everything basically the 1st day possible whether AMCAS or secondary (OK a few I got lazy on but you get the picture). Don't underestimate the value of prewriting and submitting early. This allowed me to get early IIs which basically allowed for more leeway in terms of rescheduling/grouping interviews. If you get late IIs, you have a smaller chance of being able to group them because the available dates are more and more limited.
2) Because I had so many IIs, I could group them into bunches. So I would fly in for 2 weeks, do 10 interviews back to back, and repeat the process. Most schools are actually surprisingly receptive to accommodating your schedule when you're abroad on something fantastic like a Fulbright (research one is the one with the click, ETA not so much but still great experience). They really want you (assuming other parts of the app are on par)
2a) Yes, grouping interviews can be very draining and requires some endurance. The end of one of my series was from East to West to East and ended up taking a red eye and going straight to the interview from the airport. Really try to research and plan out your interviews before scheduling them because many do not like or won't allow you to change. Try to group interviews in similar locations as much as possible to both save your endurance, money, and sanity. That being said, you will likely have to suck it up and make some sacrifices like sleeping in your suit on the plane and zombie walking into an interview and laughing it off with an ice breaker about how wonderful your travel has been. Also, you may not be able to schedule your "easier" or less important interviews first.
2b) Figure out the regulations of your program and plan in advance. If your program guidelines are incompatible with what you expect, then talk to the program in advance. Let them know and they will usually try to accommodate.
3) Enroll in TSA Precheck, rewards credit/debit card, hotel points program, and a frequent flier program. These go a LONG way. Especially when the last flight to Cali from Boston leaves at X pm and you leave the interview late, you want to breeze through the TSA Precheck line and catch that last flight. Stay with hosts as much as possible unless that's not your thing.
4) Don't worry about not attending stuff like the dinners the night before. I couldn't make all but a few of them because of my schedule and everything turned out fine! Schools know your schedules can be tight.
5) Send In the Area Emails if you have other schools in an area that you received an II from. May or may not work but doesn't hurt to try. One guy I met on the interview trail was also abroad the entire year and he walked into a school's admissions office and begged them for an interview because he had another one in the city and he ended up getting a call the morning of an interview when someone dropped out and he hopped into his suit and rushed over.
6) If you are financially restricted, plan everything twice (maybe even some educated hypotheticals) and double check before the cycle starts (aka right now). International trips like this on top of applying can be a huge drain on your savings. Bottom line, this is an investment and likely chump change compared to the debt you will accrue going to med school unless you get lucky. Also, think carefully about your options. if first class is maybe $100-200 more and you have a red eye and need to go straight to your #1 school interview, maybe it's worth it (just a cost as if you stayed in a hotel) to be a bit more rested.
In terms of interviewing early, it is much more important for schools that are rolling or semi-rolling. The number of slots at these schools will decrease over time ...For schools that aren't rolling, you really don't have any advantage in the final decision for interviewing early. You may have a slight advantage in getting an interview if you submit an app early, however. Look up UMich's webpage/twitter; they have a PPT of how their IIs/Acceptances vary by month of AMCAS submission. You will see that it is clearly skewed towards early applicants whatever the reason may be.
Yes, some schools ask in their secondary about restrictions on your interview. Some schools (maybe like 1-2) like Yale do offer Skype but I really think in-person is the way to go. With most schools, you have to wait until you get an II and then talk to the office directly about your situation.
Edit: Looking back on my cycle and the way I handled it, I would only change on thing. If you do end up getting tons of IIs, you might consider only attending your top choices if you want to take a bit of a gamble. Obviously you'd be a strong candidate if you get that many IIs.