If my opponents play an Elder Dragon to his base, and I play Star Crossed to send it back to his hand, does the 1 damage play effect from Elder still resolve? Or does it not since he goes back to hand first?

Yes, the 1 damage from Elder Dragon's "When you play me" ability will still resolve.

According to RiftJudge FAQ #10299, when you play a unit with a "When you play me" (WYPM) ability, the unit enters the battlefield, and its triggered ability is placed on the chain. This creates a Closed State, allowing you to play Star-Crossed.

Sequence of Events:

1. Opponent plays Elder Dragon: It enters the battlefield. Its "When you play me" triggered ability is placed on the chain. This creates a Closed State.
2. Your Reaction: You play Star-Crossed as a [Reaction], adding it to the top of the chain, targeting your own friendly unit and the Elder Dragon.
3. Resolution (LIFO - Last-In, First-Out):
   • Star-Crossed resolves first: It returns both targeted units (your friendly unit and the Elder Dragon) to their owners' hands.
   • Elder Dragon's "When you play me" ability resolves second: Because the ability is already on the chain, it proceeds to resolve. The ability attempts to resolve against its targets. Since the Elder Dragon is no longer on the battlefield, it is treated as having "null" attributes (Rule 359.3.e.12), but the damage it was instructed to deal to the enemy units still executes.

Summary: The Elder Dragon's triggered ability does not "fizzle" just because the unit itself left the battlefield. The ability continues to resolve, and the 1 damage will still be dealt to the targets chosen when the ability was originally placed on the chain.