My opponent casts Promising Future, banishing Elder Dragon. I banish Fizz. At which point do I choose the spell Fizz replays? When this happens, can I target his Elder with a Star-Crossed replayed from Fizz?

Yes on both counts. You pick the trash spell when Fizz's "when you play me" ability is finalized onto the chain — not when Fizz hits the board — and by that point Elder Dragon is already in play, so Star-Crossed can legally choose it.

Here's the flow. Since you're the next player after the Promising Future caster, you play Fizz first and your opponent plays Elder Dragon second, both during Promising Future's resolution. Play-effect triggers can't be finalized while a spell is resolving, so both "when you play me" abilities wait until Promising Future finishes, then get placed on the chain — turn player first, so Elder Dragon's trigger goes down first and Fizz's sits on top (Rule 383.3.d.1). Fizz's ability therefore resolves first, and targeting for the spell it replays is declared as the trigger is finalized and played (FAQs #6132 / #10020), with Star-Crossed's own targets (a friendly unit plus an enemy unit) chosen when you play it — Elder Dragon is a legal enemy unit at that moment. You still pay Star-Crossed's 1 Power.

Star-Crossed then finalizes on top of Elder Dragon's trigger, resolves, and bounces both units to hand; Elder Dragon's trigger later resolves with an illegal target (changed to a non-board zone) and does nothing (Rule 359.3.e.5). Note this only works because you win the trigger ordering — if Elder Dragon's trigger resolved first, its 1 damage would be lethal to Fizz thanks to its passive.