Can Shadow (vex signature unità) be played on Battlefield controlled by Oppo? Or only in base and ally Battlefield?

Shadow doesn't have action speed, it's his ability that's action speed

No — Shadow can only be played to your Base or to a battlefield **you** control, never to one your opponent controls. Shadow is a Signature unit with no Ambush or other permission to bypass location rules, so it obeys the normal restriction: valid play locations are your Base and battlefields you control (FAQ #6989). Attacking an opponent's battlefield doesn't make it yours — it only becomes Contested, and Contested battlefields you don't control are still invalid locations (Rule 190.3.a.1). Same for [Action] timing: that keyword only widens *when* you may activate his ability (including during Showdowns), not *where* (but he still has to be on board and you have to be able to pay the ability's cost for you to even activate it). So Shadow's "If you play me to a battlefield, I enter ready" realistically fires when you drop it onto a battlefield you control — most often your own attacking… sorry, defending one, or a rear battlefield — and that's also the only way its Stun ability ("an enemy unit attacking here") will have anything to target.