If a user plays Star Crossed as a reaction to a unit played triggered ability, does the ability still resolve? And if, like, I move to a battlefield with Diana Lunari and trigger its ability and opponent plays Star Crossed, will I still do the predict thing even if the card is in my hand?

Yes to both.

1. Star-Crossed in reaction to a "when you play me" triggered ability — the ability still resolves. Once a triggered ability is on the chain, it is independent of its source. Returning the unit to hand via Star-Crossed does not stop the ability from resolving. (FAQ #10679, #10299; Rule 392.)

2. Diana, Lunari's ability after being Star-Crossed — the Predict still happens even if Diana is back in your hand. Her trigger ("when a showdown begins here") already fired and is on the chain. The effect — pay [1], Predict, reveal the top card, draw if it's a spell — does not require Diana to be on the board when it resolves. It operates entirely on your Main Deck. So yes, you still Predict and potentially draw. (FAQ #10222, #9879, #10225.)