how to show all spin groups are double covers?

Question

If that's the definition, then how do we know double covers of $SO(n)$ exist?

Answer 1

This is a consequence of covering theory. Its central theorem says that for a reasonable topological space (like the manifold $\operatorname{SO}(n)$) path-connected coverings correspond in a certain way to subgroups of the fundamental group, and the degree of such a covering corresponds to the index of the subgroup.
The fundamental group of $\operatorname{SO}(n)$ is $\mathbb Z/2\mathbb Z$ hence the trivial group is an index 2 subgroup. The mentioned theorem now implies the existence of a (essentially unique) path-connected double covering.