If I have a Stellacorn and Zhonya's Hidden on one battlefield, and I have Irelia and a Tideturner Hidden on another, if an opponent starts a showdown on the first battlefield, can I Ride the Wind my Irelia into the other battlefield while activating Tideturner to move my Stellacorn while keeping my Zhonya's Hidden?

Yes — the whole line is legal, assuming Stellacorn and Irelia are actually in play (not hidden).

During the showdown at BF A, Ride the Wind is an Action, so it's playable in the Showdown Open State — it moves Irelia from B to A and readies her (joining the ongoing showdown is fine; the "no moving into combat" restriction only covers combat you're not in). Then you can play Tideturner from hidden at B — it enters at B, and its "when you play me" trigger may target Stellacorn at A because Tideturner is the explicit exception to Hidden's same-battlefield targeting (Rule 811.1.d.2). Resolution: Tideturner → A, Stellacorn → B, and Stellacorn's "when I move, draw 1" fires. Zhonya's stays facedown the whole time — nothing forces it to flip.

Two catches:

• You can't have two hidden cards on one battlefield. Only one facedown card can be hidden per battlefield (Rule 811.1.b), so "Stellacorn and Zhonya's hidden at A" (and "Irelia and Tideturner hidden at B") is an illegal setup. Stellacorn and Irelia must be in play — Tideturner can't swap with a facedown card that isn't a unit yet.
• Sequencing: each play opens its own chain, so you need two windows — either two Focus turns in the Showdown Open State, or play Tideturner as a Reaction while something is on the chain. And if you end the showdown with no units at A, you lose control and the hidden Zhonya's goes to trash — you keep it only while you still hold A.