Is there any difference between "3-dimension space" and "3-dimensional space"? We should use "3-dimension space" or "3-dimensional space" to mean a space with three dimension? Why?

The answer is 3-dimensional space. There's no grammatical reason I can give you. It's just due to common usage.

This is a really good question.

There are others:

3-bedroom house or 3-bedroomed house
3-leg stool or 3-legged stool

Sometimes the adjective wins and sometimes the noun (playing the role of of an adjective) wins. It's a usage question not a grammar question.

Sorry for such a disappointing answer.


More on compound adjectives

Thank you.