Dani;
"von Hengst: That would be telling.
1) It could be one of a number of reasons: you could be of a bastard royal line, or a royal ancestor might have married a commoner (possibly even of the Heroic line) without the correct dispensation, or one's original estates and titles might have been lost in battle over the centuries.
Since none of those backgrounds imply any kind of stigma or opprobrium, though, we think it's fair to say a character with that particular combination to be considered very nearly noble (and eligible for a range of noble marriages) - and you probably do have a reasonably large estate somewhere, but as a private holding rather than one over which you'd exercise what we're gonna casually call feudal rights. (I.e., you can't thrash the peasants with quite the same degree of legal impunity as Durand's family can.)
2) And, yes, you can go from destitution from the aristocracy - particularly with the kind of magical education you're getting at the school. It's not at all unheard of for very powerful wizards to receive titles and holdings from various duchesses, dukes and princelings if one joins their courts - and, even if you just stay in Mineta, the Captain (and, for that matter, families like the Chards) can pull strings and confer nobility in exchange for unusual service.
3) Oh, I'm sorry, but our hypothetical plans for imperial politics are super-classified.
1. Year 3 is when you will see the majority of the specializations begin to appear, but some will show up in Y2.
2. The tradition is that there's a "Dux Alumni," or leader of the students, who delivers a valedictory address at the end of their senior year. This person doesn't necessarily take more courses than anybody else, if that's the question; grades are expected to be good (and the DA typically is academically at the top of their class), but intangibles are considered as well. Extra-curricular activities, prominent triumphs, "enriching" the Academy or Mineta more generally - all of those can play a part. (As can professorial favoritism, though that really shouldn't be the case.)
And, in some cases, there really have been clear leaders of a given class: a century or so back a kid named Josaia Bloom, though a middle-of-the-road student, was credited with organizing the student body's defense against a siege from the Harpy Queen and her brood, and his leadership and clear-headedness under pressure left such an impression that nobody else could really have been appointed to the job.
(His actual valedictory speech was, from all accounts, a mumbled disaster, but he's still remembered fondly.)
3. Thei: Time Slot.
1) If we understand the question correctly - you are asking if a character who's of royal background and descended from heroes could have a hereditary knighthood attached to the family name... then yes, it's defensible. You wouldn't be in possession of anything resembling a feudal estate, your family wouldn't have any vassals, and you certainly wouldn't enjoy tax exemptions or class-based legal protections (unless your legal skills were incredibly high and you wanted to push your luck in court), but if you tacked an honorific onto your name in your official correspondence you could probably get away with it.
2) Actually "divine bloodline without royal descent" could probably be a background in and of itself.
There are Heroic lines that trace back to Auncish royals, and if you choose that background and consider that it applies to you, it's totally legit.
3 and 4) Travel times for summer vacation can and do vary widely, and wealth does factor into it to a degree. If you're from a faraway part of Elumia, expect some time on an air ship.
Travel is different from what you have seen, systems-wise, in Y1.