Do I need a SKU for every product, and how do variants get theirs?

Yes, for Simple and Variable products. The SKU field is in the product's Inventory card, and it must be unique in your store.

• As you type, WCart checks it: ✓ Available or Already in use.
• If the field is empty, click Use "…" to fill in a code built from the product's URL slug.
• For a Variable product, this is the SKU (parent). Give each variant its own code in the SKU column of the variants table.

Affiliate, Downloadable and Bundle products don't need a SKU.

The SKU is also how stock is tracked by location. Manage stock in the Product Inventory card asks for a SKU first.