Can one prize need a minimum spend, or work only on some products?

Yes. Make an ordinary WooCommerce coupon with the limits you want (minimum or maximum spend, products or categories, excluded ones, a cap on items, sale items, individual use), then choose it in that prize's Coupon rules field on the Wheel screen. The winner's new code copies those limits; the prize still sets the discount, and the email lock, usage limit and expiry still come from the Settings screen, unless you set Expires after (days) beside the field. The winner is shown the conditions with their code and in the email. A Fixed product discount prize without coupon rules takes its amount off every item in the cart, and the Wheel screen warns you about that.