I think 父ちゃん奮闘だぁ! is a quote from the game (it was in the demo, if I remember well), Sazh refers to himself as 父ちゃん when he speaks to his chocobo. It was something like "Daddy has to fight!", although we might have to wait until the US version to check the actual quote.
Otherwise, I would actually use "Pulse" instead of "World Below", since they read 下界 as "Pulse" in Japanese... Besides, in one of the English trailers, the characters say "Pulse's fal'Cie" (下界のファルシ).
Great work on the translation, anyway!
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