There is a zero index for a scalar. A rank-1 tensor is a vector. A vector does change under rotation. The number of numbers needed is the number of dimensions in the vector space raised by the rank. Symmetry can often simplify the number of numbers actually needed to describe a tensor.
Physics equations involve tensors of the same rank.
There are scalar equations, polar vector equations, axial vector equations, and equations for higher rank tensors. Since the same rank tensors are on both sides, the identity is preserved under a rotational transformation. One could decide to arbitrarily combine tensor equations of different rank, and they would still be valid under the transformation. There are ways to switch ranks. If there are two vectors and one wants a result that is a scalar, that requires the intervention of a metric to broker the transaction.