All that is said, is that he was sent East to Lyon's, it's speculated that it was done so in order for him to avoid the Brotherhood infighting (probably the Circle of Steel) and the NCR War. There is no mention of how he crossed, which leaves this open to speculation. And if it happened years ago, they are probably more worried about the current situation with the Enclave and the Purifier than an airship.
His father was not a High Elder of the BOS but a paladin when he died, leaving Arthur the last Maxson alive. So while yes the Maxson's have been leaders, they've been groomed for the position and thus they take it when the previous Maxson dies or they are deemed ready by the ruling council of the West Coast BOS to take up the position of High Elder. As for their internal policies, we actually do know how promotions work, at least for West Coast BOS; you do your job competently, you get promoted eventually; you find an amazing tech find, you get a promotion; you show an extreme aptitude for sciences and leadership, and go above the call of duty, you get promoted. It's a meritocracy. If it wasn't, than Jonathan Maxson, Arthur's father, should've been a High Elder on his death, rather than a paladin.