Gangplank Naval, if I use Starcrossed, can they use their ability? Will they get +3 and do they need to pull back to hand? Or can they stay at the same battlefield? It says "or put to hand," so you don't need to, right?

Yes — if Gangplank is already Empowered, he stays on the board and gets +3 Shield this turn instead of being bounced.

His Empowered ability is a mandatory replacement effect: if a spell that chooses him would return him to hand, that doesn't happen — he gets +3 [S] (Shield 3, i.e., +3 Might while defending) for the turn instead (Rule 814, FAQ #11914). Star-Crossed does choose him (choosing = targeting), so the replacement applies automatically. The Gangplank player doesn't activate anything — it just happens, and there's no choice to "use" it or decline it.

To be clear: Star-Crossed has no "or put to hand" option — it always returns both chosen units to hand. It's Gangplank's ability that swaps the bounce for the +3, so he stays right where he is. The other unit Star-Crossed chose still returns to hand as normal. One caveat: if he's not Empowered yet (or if Star-Crossed resolves before an Empower on the chain does), he bounces like any other unit.