The rank of a matrix is the dimension of the image of the linear map it represent. Since the only vector space of dimension 0 is the vector space denoted 0 consisting of only one elements (namely, 0), to say that a matrix is of rank 0 is to say that the image of the linear map it represents is the vector space 0.
You can convince yourself that, in turn, this implies that the matrix is the matrix 0 (the matrix having 0 in all its entries...)
Yes, the rank of a matrix is the dimension of the row space.
If a matrix has a zero-dimensional row space, it consists of a single vector - the zero vector. The space consisting of the zero vector only has dimension zero.
If a vector had an entry besides 0, then that row would not be the zero vector. Then the row space would include inifinitely many vectors corresponding to all scalar multiples of that vector... and have dimension at least one.