When Red Akali moves into a battlefield where Blue Teemo is defending, does Teemo's "When I defend" go on the chain first, or can Akali instantly use her Legend ability and move Red Akali unit back as an action?

No — Teemo's "When I defend" goes on the Chain first; you can't use the legend ability to preempt it.

Here's the order: Akali's "When I move" resolves first (dealing 1, or 2 if Empowered — if that kills Teemo, there's no Showdown at all and no defend trigger). After that chain empties, a Cleanup stages the Showdown: Teemo gains the Defender designation and his "When I defend" is placed on the Chain immediately (Rule 383.3) — there's no priority window in between.

While that trigger is on the Chain the state is Closed, and only Reaction-speed effects can be played. The Rogue Assassin legend ability is [Action], so it can't be used as a response (Rules 145.2, 381). Your first chance to move Akali back is after Teemo's trigger resolves and the Chain empties back to Showdown Open State — at that point you can activate the legend ability, move Akali to base, and the Showdown unstages before combat damage (FAQ #11755).