When I play a gear with Jarvan IV, Exemplar (legend), and the gear is Portable Portal, and I choose to set up the triggers so Portable Portal is on the chain first, then Jarvan, I use Jarvan's legend ability to move a unit and start a showdown. Does the Portable Portal trigger to ready a unit stay on the chain until after the full combat resolves, or does it resolve before combat?

No — Portable Portal's ready trigger resolves before combat, not after it. Combat can only open once the Chain is empty (Rule 462), and both the Portal's play trigger and Jarvan's move trigger are sitting on that Chain from the same event, so they both resolve first; your ordering only decides which resolves first (Jarvan's move, since it went on last, then Portal's ready), and only after the Chain empties does the Cleanup stage the Showdown/Combat at the now-contested battlefield (Rules 323.8, 323.12). No chain item ever carries over into a Showdown.

One thing the ordering can't buy you: the Portal's trigger has to pick its unit and pay the [C] when it finalizes on the Chain, which is before Jarvan's move ever resolves — so a unit that only arrives via Jarvan is not a legal choice for the ready (FAQ #13018). It can only ready a friendly unit that was already at that battlefield when the Portal was played.