I initiate a showdown with Rek'Sai, my opponent plays Star-Crossed. Do I still trigger the Rek'Sai ability or no?

Yes, Rek'Sai, Swarm Queen’s ability still triggers and resolves even if she is returned to your hand, but you cannot play the unit "here" because the location is no longer valid.

1. Initiating Combat When you move Rek'Sai to an occupied enemy battlefield, you initiate a combat showdown. Rek'Sai gains the Attacker designation, which triggers her ability: "When I attack, you may reveal the top 2 cards of your Main Deck. You may banish one, then play it. If it is a unit, you may play it here. Recycle the rest."

2. The Chain The triggered ability is placed on the chain. During the subsequent [Reaction] window, your opponent plays Star-Crossed, which adds to the chain and returns Rek'Sai to your hand.

3. Resolution (LIFO) The chain resolves in Last-In, First-Out order:

• Star-Crossed resolves first: Rek'Sai is returned to your hand.
• Rek’Sai’s ability resolves second: Abilities exist independently of their source. Even though Rek'Sai has left the battlefield, her ability still resolves. You execute the instructions to reveal the top 2 cards of your Main Deck and banish one.

4. Evaluating "Here" (Rule 359.3.f.2) Per Rule 359.3.f.2 (Referents), the term "here" is checked upon the execution of the instruction. Because Rek'Sai is now in your hand (a non-board zone) when the ability resolves, "here" no longer references the battlefield where combat was initiated. Consequently, you cannot use the permission to play the unit to that location.

Note: I could not fully verify this ruling against the retrieved rules.