Does every vector space contain a zero vector?

Yes, and yes, you are correct.

The existence of a zero vector is in fact part of the definition of what a vector space is.

Every vector space, and hence, every subspace of a vector space, contains the zero vector (by definition), and every subspace therefore has at least one subspace:

The subspace containing only the zero vector vacuously satisfies all the properties required of a subspace. It is closed under vector addition (with itself), and it is closed under scalar multiplication: any scalar times the zero vector is the zero vector.

Every vector space also contains itself, and is its own subspace.

Except in one very special situation (which one?) it follows from this that almost every vector space has two subspaces.