Irresistible Faefolk

You — the Irresistible Faefolk player — are the attacker.

When Irresistible Faefolk moves to an open (uncontrolled) battlefield, that move applies the contested status to the battlefield.^{[450][190.3.a]} Irresistible Faefolk's trigger resolves after the cleanup caused by its own move, moving the enemy unit to that battlefield. During the cleanup following the enemy unit's move, combat is staged because opposing units are now present at the contested battlefield.^{[319.8][323.9]} When combat opens, the attacker is defined as the player whose unit(s) applied the contested status.^[464.2.c.1] That is you — the player who moved Irresistible Faefolk. The opponent whose unit was pulled in is the defender.^[464.2.c.2]

No — playing a unit to a battlefield is not a move, so the ability does not trigger.

A move is defined as a permanent changing its position from one location on the board to another.^[446.1] Playing a card involves it entering the board from a non-board zone (such as your hand), which is a zone change, not a move.^[446.2] Because Irresistible Faefolk's ability specifically triggers when it moves to a battlefield, playing it directly there does not satisfy that trigger condition.

Yes — the enemy unit still moves to that battlefield.

Irresistible Faefolk's trigger condition is moving to a battlefield, and "that battlefield" in the effect is a referent to the battlefield named in that trigger condition. Information referenced from a trigger condition is checked when the trigger condition is fulfilled, not when the ability resolves.^[359.3.f.3] That means "that battlefield" was already fixed the moment Irresistible Faefolk moved there and placed the ability on the chain.

Suppose an opponent responds to the triggered ability by playing Gust (or similarly, Star-Crossed), targeting Irresistible Faefolk itself to return it to your hand. Irresistible Faefolk returns to your hand, but the triggered ability is unaffected — it's already an independent item on the chain, and the battlefield it references was fixed at trigger time. When it resolves, it moves an enemy unit to that battlefield, even though Irresistible Faefolk is no longer there.

This differs from information derived from an ability's source or targets, which is generally checked when the referencing instruction executes and may become unavailable before resolution.^{[359.3.f.1][359.3.f.2]}