Can Shadow be played directly to any battlefield (open or controlled by an opponent)? Or can It only be played to a battlefield I already control?

No — Shadow can only be played to your Base or to a battlefield you control. Its [Action] keyword is timing permission only (Showdowns, any player's turn); Rule 806.3 explicitly says a Unit with Action still has the inherent restriction of only being played to your Base or a battlefield you control. So an open/uncontrolled battlefield or one controlled by your opponent is not a legal destination, even during a combat Showdown there.

The payoff of its text is conditional on that: "If you play me to a battlefield, I enter ready" only fires when you play it to a battlefield you already hold, which is why Shadow is good as a mid-Showdown blocker at your own battlefield but can't be flashed in to contest an enemy one. If you play it to your Base it enters exhausted. Only something like Ambush (a battlefield where you have units) would expand the destination list — Shadow has none.