How to use nouns as a category?

They are, in general, not equivalent.

1 could occur in a simple story in which the reader is asked to imagine a scene. A cat runs faster than a dog. The dog tires. The cat escapes.

2 refers to a specific cat and a specific dog, both defined or implied by previous context. The statement might refer to particular animals (Mitzi the cat and Fido the dog). It might also apply to the species (the dog, the cat, the cow, the horse etc). In this latter case 2 is equivalent to 3.

3 is a general statement that applies to all cats and dogs