what are the steps of a chain in and outside of a showdown

Same four steps either way — the FEPR loop (Finalize → Execute → Pass → Resolve, Rules 337–340). Only the permissions and what happens when the chain empties differ.

Finalize: each controller completes the play steps for their Pending items in the order they were added; units, gear, and Add-resource abilities resolve immediately on finalizing (337.2). Execute: the player with Priority (in a showdown, the player with Focus) plays a legally timed card/ability or passes — in a Closed State outside a showdown that means Reaction only, while in a Showdown Closed state an Action card that opened the chain is also fine (338.1.a.2, 358.4). Pass: if everyone passes in sequence without adding anything, go to Resolve; otherwise priority moves to the next player in turn order. Resolve: the newest item resolves fully (LIFO), then back to Finalize if the chain isn't empty, or the turn returns to an Open State if it is.

The real divergence is after that last resolution: during a showdown, Focus passes to the next player in turn order (340.2.a) — except when the chain was opened by a triggered ability or an Add ability, in which case the opener keeps Focus (346.1). Outside a showdown there's no Focus at all; the turn simply returns to Neutral Open and, after any cleanup, the Turn Player gets priority back in their Main Phase.