Does zero vector have zero dimension?

Yes but here's a minor nit pick: A vector doesn't have a dimension, you want to say that the subspace spanned by the zero vector has dimension zero.

For the second question you may use the rank nullity theorem. You have $\mathrm{dim} \mathrm{ker} A = 0$ and hence $\mathrm{dim} \mathrm{im} A = \mathrm{dim}R^n - \mathrm{dim} \mathrm{ker} A = \mathrm{dim}R^n$ hence the $v_i$ span $R^n$.

The concept of dimension is applied to sets of vectors, in particular subsets of vector spaces that are also subspaces. Thus, it more appropriate to say that the subspace consisting of the zero vector has dimension zero.

If you are assuming that $A$ is a square matrix and has independent columns, it has maximal rank. A matrix with maximal rank is invertible and the linear map induced by left multiplication by $A$ is an isomorphism. Isomorphisms always map a basis to another basis.