Can I nest a category within a category?

No, unfortunately a category cannot have another category assigned to it or within it.

Can a product belong to more than one category?

Yes, on the Advanced Settings screen of the product you are editing you can assign multiple categories to the product.

Does a product have to have a category assigned?

No, no product is required to be assigned to a category.

How does a category differ from a product attribute?

A category is a broad categorization of products. While product attributes can also fall into this classification, product attributes are more intended to describe features or attributes of various products. 

An example is with Rafting activities. You might categorize your rafting activities under a "Rafting" category (to differentiate them from Kayak or Paddleboard activities), but then may assign things like difficulty levels (Category 1-5), seasons (best in the spring for winter run-off), or things like "Adults Only Experience" to your activities as various product attributes.