Vex, Apathetic

No. Vex's trigger condition identifies the unit that was just played, and the later word "it" refers back to that information.^{383.2.a359.3.f.3} Vex's controller never chooses among units, so the unit is not a target.^{355.10.d355.10.d.1}

An ability that prevents enemy spells or abilities from choosing the unit therefore does not stop Vex from stunning it.⁷⁵⁷⁷⁵⁸ [Deflect] also adds no cost because Vex's ability does not choose the unit.^809.1.d

No, Vex's ability does not trigger or stun an opponent's gear with [Deploy] at a battlefield when the opponent's Lady of Clockwork makes it a unit, even while Vex is at a battlefield.

A card is played only when it finishes the process of playing a card in its entirety.^350350.1 Lady of Clockwork's ability changes the chosen gear's card type at its current location instead of playing it. Because no unit was played, Vex's trigger condition is not met.^383.2.a383.3

Yes, you can move the unit because Vex prevents the opponent identified by its trigger from moving it that turn, not every player. Vex's trigger condition identifies both the opponent who played the unit and the unit itself, so the later words "they" and "it" refer back to that information.^{383.2.a359.3.f.3}

Charm, Void Assault, and Moonfall each instruct you to move an enemy unit. The player who performs a game action is responsible for it, even when another player controls the affected unit.^412.1 In each case, you perform the move, not the opponent who controls the unit, so Vex's restriction does not apply.