When I go to fulfill an order, why does it take me to the "Fulfill Orders" screen instead of showing me the screen like I'm about to create an item fulfillment record? Also, how do I get NetSuite to not do that?

Short answer - you can't. Longer answer - "Fulfill Orders" page is a mass process. If you want to force people to fill out specific fields on item fulfillment remove the associated permission from their roles and leave the create/edit capability on fulfillment transactions. Then they must load the Sales Order in order to fulfill it, which leads to the screen you expect to see.