Общий - Did they build a small house last year?
Специальные : When did they build a small house? How did they build a small house last year? What did they build last year? Where did they build a small house last year?
Альтернативный - Did they build a small house or a big cottage last year?
Разделительный (вопрос с хвостиком) They built a small house last year, didn't they?
Вопрос к подлежащему - Who built a small house last year?