Absolutely he would get leave. The time depends a little on the circumstances, how critical he is in his role, how close he was with the nephew etc. Probably a week, but could be more or less depending on those factors.
Padres have the power to make that sort of thing happen, but he might also go to his immediate superior with the request depending on how religious he was and his relationship with the superior.
It depends.
No really, it does depend.
In the US (at least when I was active duty) Emergency Leave for a death in the family would be limited to 'immediate family' members. A nephew doesn't count as immediate, unless the service member can prove that he was acting as the nephew's primary care-giver. For example, when my father-in-law was diagnosed with grade 4 glioblastoma, I wasn't eligible for Emergency Leave, because he wasn't immediate family. Chaplains can strongly recommend for emergency leave, but they can't grant it.
Of course, he could always ask for leave, and generally it would be granted depending (there's that word again) on what his duties and responsibilities are. If he's a critical point of failure for anything remotely important, then he'd have to get a replacement.
Getting back to the States would be easy. The Air Force ran cargo flights to/from Japan daily, and he'd be able to fly what's called 'Space A' or Space Avaialble. If his command cut him orders, he could get a seat faster.
Suffice it to say, if you want him to get home, or not get home, there are valid military reasons for both.