Learn Spanish (Latin America): El perro está fuera de la casa.

No, you need the de. In Spanish, many positional words like fuera (outside), dentro (inside), or al lado (next to) need de when they are followed by the specific place. It's helpful to think of it as literally saying "outside of the house."

I sometimes hear the word afuera. Could I say "afuera de la casa" instead?

Yes! While fuera de is the traditional preposition, in Latin American Spanish it is very common to hear afuera de used in the exact same way. Both mean "outside of the house" and will be perfectly understood.

Why do we use está here instead of es?

We use está (from the verb estar) because the sentence is talking about physical location. Spanish always uses estar to state where someone or something is located, regardless of whether that location is temporary or permanent.

What happens to de la if the dog is outside a masculine place, like "the building" (el edificio)?