No it does not. Like every other windfall world, it only produces a good when you use a "produce a good on a windfall" power. This can either be a power from a card (e.g., Expanding Colony or Terraforming Guild), or the bonus you get for calling Produce (which allows you to produce a good on one windfall world).
Correct. Rebel Sympathizers' Produce power acts just like Runaway Robot's Produce power in the base game, as described on page 11 of the base game rules "Draw on Produced Windfall".