how Bone Skewer resolves in the chain alone. And how it works in a chain with reposte?

Bone Skewer alone on the chain: it resolves fully, then the unit it forces out enters, gets stunned, and only then — once the chain is empty — can any Showdown begin. With Riposte in response: Riposte resolves first (newest-first, Rule 340.1), counters Bone Skewer, and none of Bone Skewer's effects happen at all.

Bone Skewer alone: On resolution you choose a battlefield, the opponent reveals their hand, and if you pick a unit, they play it there ignoring costs. That play puts the unit on the Chain as a Pending item, and "If they do, then do this: [Stun] it" is placed on the Chain after it (per the current errata text, this is the linked reflexive instruction). After Bone Skewer finishes, the unit finalizes and enters the battlefield exhausted — applying Contested if it lands on a battlefield you control or have units on. The Showdown can't begin yet because the Chain isn't empty and the turn isn't in a Neutral Open State (Rule 344; FAQ #10033). The stun trigger then resolves, so the unit is already stunned (0 Might in combat, Rule 423.1.b) before the Showdown initiates. When it does, if your units are there it's a Combat Showdown and the opponent — whose unit applied Contested — is the Attacker and gains Focus (FAQ #11243). If you pointed it at an empty battlefield, it becomes a Non-Combat Showdown the opponent wins, effectively handing them the Conquer.

With Riposte: Riposte has [Reaction], so it can be played while Bone Skewer sits on the Chain. It resolves first and "Counter that spell" negates Bone Skewer: it does nothing, is cleared from the Chain to the trash, and the already-paid costs aren't refunded (Rule 425.1.a–c). No battlefield is chosen, no hand is revealed, no unit is played, and no Showdown is staged. The friendly unit does get +[S] equal to Bone Skewer's Energy cost this turn — that's +2 [S].