Zinc ore must be mined with an iron pickaxe or higher, or else it drops nothing. Zinc ore always drops itself, regardless of whether the player used Silk Touch.
By far the most effective way to process Zinc ore is first use a pair of Crushing Wheels, by doing this you can have an extra 30% chance to get 2 more crushed Zinc instead of the usual 1, then use Bulk Washing on the Crushed Zinc to get 10 Zinc Nuggets per Crushed Zinc and a chance to get 5 more Nuggets, then craft the nuggets into ingots with Auto Packing or crafting.