i thing that below is to describe something with much more distance

Not necessary, But in real life, yes. 狗在椅子下边 / 狗在椅子下面 is more natural.

Omitting the last word is a way to shorten sentences in Chinese writing. In ancient Chinese scriptures, the meanings of the sentences can be concentrated into even less words, but each word may have more meanings. In this case, an ancient way of saying it may be 犬於座下 (dog - at - chair - under) haha