If Irresistible Faefolk move to BF and targeting opp unit in base, and opp react with Gust targeting Faefolk in BF, does the unit in base still move to BF?

Yes, the enemy unit in base still moves to the battlefield.

Gust removes Irresistible Faefolk, but its triggered ability is already on the chain. Per Rule 359.3.f.3, the ability's reference to "that battlefield" (where the Faefolk arrived) was locked in when the trigger condition was fulfilled. The trigger doesn't need the Faefolk to still be on the board to resolve — it only needs a valid target for its effect (the enemy unit in base), which remains unaffected by Gust.