Yeah, the term existed way before Pokémon. Translation-wise it was definitely a pokemon reference. No other official translation uses "elite four" afaik.
Yeah,i don't miss that at all. I guess i never understand the appeal of that ship when Sensei seem to care so little about Eiko and don't strike me with the coat. It's just because Sensei know she will see Eiko again since they're in the same school.
I love that Enami let her keep the coat now, it shows how much they have progressed together compared to before she was her student. And congratulating her months after is such an Enami thing to do.